Enter the coefficients a, b and c of the quadratic function y = ax² + bx + c. The equation below is linked to the input fields, so you can also edit the coefficients in the equation directly.
Enter numbers only. Decimals, negative numbers and fractions such as 3/4 are OK. A blank a counts as 1 (x² is the same as 1x²), and a blank b or c counts as 0.
Result and graph
Enter the coefficients a, b and c in the fields on the left and press "Calculate". The steps for completing the square, the vertex and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and graphs
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter the coefficients, and the quadratic function \(y = ax^{2} + bx + c\) is rewritten in the form \(y = a(x - p)^{2} + q\) (vertex form) by completing the square
- The steps follow the textbook method line by line: factor out the leading coefficient, add and subtract the square of half the \(x\) coefficient, then write the perfect square
- You also get the vertex \((p,\ q)\), the axis of symmetry \(x = p\), which way the graph opens (upward or downward) and the maximum or minimum value, along with a graph of the parabola
- Answers are shown both as reduced fractions (exact values) such as \(-\dfrac{5}{4}\) and as decimals. Coefficients can be decimals, negative numbers or fractions such as 3/4
- The calculator shows the equation the way you would write it by hand, and you can edit the coefficients right in the equation
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are also on this page
This page is for quadratic functions (the x² coefficient a is not 0). To solve an equation of the form ax²+bx+c = 0, use the Quadratic Formula Calculator in the related tools.
What is this calculation used for?
Finding the highest point of a ball or a fountain (physics and sports)
The height of a ball thrown upward, or of the water in a fountain, is often modeled as a quadratic function of time. For example, if the height is \(h = -16t^{2} + 64t\) (in feet, with \(t\) in seconds, ignoring air resistance), completing the square gives \(h = -16(t - 2)^{2} + 64\). You can read off that the ball reaches its highest point, 64 feet, after 2 seconds.
Pulling the vertex, "the highest point and when it happens", straight out of the equation is the strength of completing the square, and projectile problems in physics are a classic use of it.
Finding the price that maximizes profit (business and economics)
When "raising the price increases the profit per item but reduces how many you sell", profit can sometimes be modeled as a quadratic function of price that opens downward. The price that gives the most profit is exactly the vertex of the parabola. For example, if the profit is \(y = -x^{2} + 60x - 500\) dollars (with \(x\) the price in dollars), the vertex \(x = 30\) is the price that maximizes profit: $30.
Real markets are more complicated, but "look for the best value at the vertex" is a starting point for thinking about pricing and how to split an ad budget.
Making the largest garden bed with a fixed length of fence (design and DIY)
For a rectangle with a fixed perimeter, the area is a quadratic function of the side length. For example, with 40 feet of fence, if the width is \(x\) feet, the length is \(20 - x\) feet, and the area is \(y = x(20 - x) = -x^{2} + 20x\). Completing the square gives \(y = -(x - 10)^{2} + 100\), so the largest area is 100 square feet when \(x = 10\) (a square).
This is the most basic form of an optimization problem: getting the most out of limited materials or a limited budget.
Designing antennas and light reflectors (engineering)
Satellite dishes and the reflectors in car headlights and flashlights have the shape of a parabola spun around its axis (a paraboloid). They use a property of the parabola: radio waves or light coming in parallel to the axis all bounce to one point on the axis (the focus), and light sent out from the focus travels straight out parallel to the axis.
In design, a parabola is described by where its vertex is and how wide it opens in which direction, so the idea of vertex form \(y = a(x - p)^{2} + q\), built around the vertex, is the foundation.
Formulas and graphs
Completing the square (converting to vertex form)
Graph
Standard notation
(the usual math form)
\(y\)
\(=\)
\(a\)
\(\left(x - p\right)^{2}\)
\(+\)
\(q\)
In words
(symbols replaced with words)
④
\(y\): value of the quadratic function
\(=\)
①
\(a\): \(x^{2}\) coefficient
②
square of \(x\) minus the vertex \(x\)-coordinate \(p\)
\(+\)
③
\(q\): \(y\)-coordinate of the vertex
The formula in words
①
Take the
\(x^{2}\) coefficient \(a\)
②
multiply it by the
square of \(x\) minus the vertex \(x\)-coordinate \(p\)
③
add the
\(y\)-coordinate of the vertex \(q\)
④
and you get the
value of the quadratic function \(y\)
(in this form, you can see the vertex \((p,\ q)\) and the axis \(x = p\) at a glance)
Quick example
Completing the square on \(y = 2x^{2} - 8x + 3\) gives \(y = 2(x - 2)^{2} - 5\) (vertex \((2,\ -5)\))
value of the function \(y\)
\(=\)
\(x^{2}\) coefficient (2)
square of \(x\) minus 2
\(+\)
\(y\)-coordinate of the vertex (−5)
\(y = 2x^{2} - 8x + 3 = 2\left(x^{2} - 4x\right) + 3\)
\(y = 2\left(x^{2} - 4x + 4 - 4\right) + 3 = 2\left\{\left(x - 2\right)^{2} - 4\right\} + 3\)
\(y = 2\left(x - 2\right)^{2} - 8 + 3 = 2\left(x - 2\right)^{2} - 5\)
Key idea
Completing the square rewrites a quadratic function as "a squared part plus a constant". Since \((x - p)^{2}\) is a square, it is always 0 or more, and it is 0 only when \(x = p\). So if \(a > 0\), \(y\) is smallest at \(x = p\) (minimum value \(q\)), and if \(a < 0\), \(y\) is largest there (maximum value \(q\)). The value of this rewrite is that you can read the vertex and the maximum or minimum from the form of the equation alone.
The form \(y = ax^{2} + bx + c\) is called standard form, and \(y = a(x - p)^{2} + q\) is called vertex form (the vertex can be seen at a glance). When you graph a quadratic or look for its maximum or minimum, converting to vertex form is the usual first move. (Many textbooks write vertex form as \(y = a(x - h)^{2} + k\), with the vertex \((h,\ k)\).)
\(x\)-coordinate of the vertex (the axis of symmetry)
Standard notation
(the usual math form)
\(p\)
\(=\)
\(-b\)
\(\div\)
\((\)
\(2a\)
\()\)
In words
(symbols replaced with words)
③
\(p\): \(x\)-coordinate of the vertex
\(=\)
①
\(-b\): \(x\) coefficient with its sign changed
\(\div\)
\((\)
②
\(2a\): twice the \(x^{2}\) coefficient
\()\)
The formula in words
①
Take the
\(x\) coefficient with its sign changed, \(-b\)
②
divide it by
twice the \(x^{2}\) coefficient, \(2a\)
③
and you get the
\(x\)-coordinate of the vertex \(p\)
(the axis of symmetry is the line \(x = p\))
Quick example
The \(x\)-coordinate of the vertex of \(y = 2x^{2} - 8x + 3\) (\(a = 2,\ b = -8\)) is
\(x\)-coordinate of the vertex \(p\)
\(=\)
\(b\) with its sign changed (8)
\(\div\)
\((\)
twice \(a\) (4)
\()\)
\(p = -\dfrac{b}{2a} = -\dfrac{-8}{2 \times 2} = \dfrac{8}{4} = 2\)
Key idea
This is the vertex formula you get by completing the square all the way with letters instead of numbers. Without rewriting the equation every time, you can find the \(x\)-coordinate of the vertex straight from \(a\) and \(b\). (This calculator also finds \(p\) with this formula.)
A parabola is symmetric about its axis (the line \(x = p\)), so \(p\) is also where the graph folds over. It has the same form as \(-\dfrac{b}{2a}\), the part before the "±" in the quadratic formula \(x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\). This shows that the axis is exactly halfway between the two solutions.
\(y\)-coordinate of the vertex (maximum or minimum value)
Graph
Standard notation
(the usual math form)
\(q\)
\(=\)
\(c\)
\(-\)
\(b^{2}\)
\(\div\)
\((\)
\(4a\)
\()\)
In words
(symbols replaced with words)
④
\(q\): \(y\)-coordinate of the vertex
\(=\)
①
\(c\): constant
\(-\)
②
\(b^{2}\): square of the \(x\) coefficient
\(\div\)
\((\)
③
\(4a\): 4 times the \(x^{2}\) coefficient
\()\)
The formula in words
①
From the
constant \(c\)
②
subtract the
square of the \(x\) coefficient \(b\)
③
divided by
4 times the \(x^{2}\) coefficient, \(4a\)
④
and you get the
\(y\)-coordinate of the vertex \(q\)
(the minimum value if \(a > 0\), the maximum value if \(a < 0\))
Quick example
The \(y\)-coordinate of the vertex of \(y = 2x^{2} - 8x + 3\) (\(a = 2,\ b = -8,\ c = 3\)) is
\(y\)-coordinate of the vertex \(q\)
\(=\)
constant (3)
\(-\)
\(b\) squared (64)
\(\div\)
\((\)
4 times \(a\) (8)
\()\)
\(q = c - \dfrac{b^{2}}{4a} = 3 - \dfrac{(-8)^{2}}{4 \times 2} = 3 - \dfrac{64}{8} = 3 - 8 = -5\)
Key idea
\(q\) is the value of \(y\) when \(x = p\), so plugging \(x = p\) into the original equation gives the same answer. This is a handy way to check your work.
A parabola that opens upward (\(a > 0\)) has its lowest point at the vertex, so \(q\) is the minimum value. A parabola that opens downward (\(a < 0\)) has its highest point at the vertex, so \(q\) is the maximum value. But when the domain (the values \(x\) is allowed to take) is limited, the vertex may fall outside it, and you need to compare with the values at the ends of the domain.
By completing the square, the quadratic function \(y = ax^{2} + bx + c\) can be rewritten in vertex form \(y = a(x - p)^{2} + q\), where \(p = -\dfrac{b}{2a}\) and \(q = c - \dfrac{b^{2}}{4a}\). The vertex is \((p,\ q)\) and the axis of symmetry is the line \(x = p\). If \(a > 0\), the minimum value is \(q\) at \(x = p\); if \(a < 0\), the maximum value is \(q\).
Symbols and terms
Symbols
| \(a,\ b,\ c\) | a, b, c | The coefficients of the quadratic function \(y = ax^{2} + bx + c\). \(a\) is the \(x^{2}\) coefficient (not 0), \(b\) is the \(x\) coefficient, and \(c\) is the constant. Letters from the start of the alphabet, \(a,\ b,\ c\), are usually used for fixed numbers. |
| \(x,\ y\) | x, y | Variables. \(x\) is the input and \(y\) is the output. Letters from the end of the alphabet, \(x,\ y,\ z\), are usually used for numbers that can take many values. |
| \(p\) | pee | The \(x\)-coordinate of the vertex, which is also where the axis of symmetry (the line \(x = p\)) is. Many US textbooks use \(h\) for this. |
| \(q\) | cue | The \(y\)-coordinate of the vertex. It is the minimum value if \(a > 0\) and the maximum value if \(a < 0\). It is the letter after \(p\), so the two are used as a pair. Many US textbooks use \(k\) for this. |
| \((x - p)^{2}\) | x minus p, squared | The squared part made by completing the square. As a square it is always 0 or more, and it is 0 only when \(x = p\). This is exactly why the maximum or minimum is at the vertex. |
| \(x^{2}\) | x squared | \(x\) multiplied by itself (\(x^{2} = x \times x\)). The small raised number (the exponent) tells how many times it is multiplied. |
Terms
| quadratic function | A function of the form \(y = ax^{2} + bx + c\) (\(a \neq 0\)). Its graph is a parabola. It is taught in Algebra 1 and Algebra 2. |
| parabola | The symmetric U-shaped curve that is the graph of a quadratic function. It is the same shape as the path of a thrown ball. |
| vertex | The turning point of a parabola: the lowest point if it opens upward, the highest point if it opens downward. Its coordinates are \((p,\ q)\). |
| axis of symmetry | The vertical line \(x = p\) that a parabola is symmetric about. The parabola is a mirror image of itself across this line, and the vertex lies on it. |
| completing the square | Rewriting \(y = ax^{2} + bx + c\) in the form \(y = a(x - p)^{2} + q\). The name comes from "making a perfect square". It is the basic tool for finding the vertex, the axis and the maximum or minimum, and the quadratic formula itself is derived with it. |
| perfect square | An expression that is something squared, such as \((x - 2)^{2} = x^{2} - 4x + 4\). A square is a number times itself: the square of \(3\) is \(3^{2} = 9\). |
| standard form | The form \(y = ax^{2} + bx + c\). It is the usual expanded form, and you can easily read where it crosses the \(y\)-axis (\(y = c\) when \(x = 0\)). |
| vertex form | The form \(y = a(x - p)^{2} + q\) (often written \(y = a(x - h)^{2} + k\)). You can see the vertex \((p,\ q)\) at a glance. Completing the square converts standard form into vertex form. |
| opens upward | The graph opens like a valley. This happens when \(a > 0\), and the vertex is the lowest point. |
| opens downward | The graph opens like a hill. This happens when \(a < 0\), and the vertex is the highest point. |
| maximum value | The largest value the function \(y\) can take. For a quadratic that opens downward, the \(y\)-coordinate of the vertex \(q\) is the maximum value. |
| minimum value | The smallest value the function \(y\) can take. For a quadratic that opens upward, the \(y\)-coordinate of the vertex \(q\) is the minimum value. |
| coefficient | The number in front of a letter. In \(2x^{2}\), 2 is the coefficient. A term with no number written, such as \(x^{2}\), has coefficient 1. |
| constant term | A number-only term without \(x\), such as the 3 in \(2x^{2} - 8x + 3\). It is called constant because it does not change whatever the value of \(x\). |
| symmetric | Matching exactly when folded over. If you fold a parabola along its axis, the left and right sides match exactly (line symmetry). |
| domain | The set of values \(x\) is allowed to take. When the domain is limited (for example, \(0 \le x \le 3\)), the maximum or minimum may be at an end of the domain instead of at the vertex. |
| translation | Sliding a figure without changing its shape (also called a shift). The graph of \(y = a(x - p)^{2} + q\) is the graph of \(y = ax^{2}\) shifted \(p\) in the \(x\) direction and \(q\) in the \(y\) direction. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back to review these topics is the fastest way forward.
| Functions and graphs (Grade 8 and Algebra 1) |
|
| Expanding and factoring (Algebra 1) |
|
| Signed numbers and algebraic expressions (Grades 6–7) |
|
| Working with fractions (Grades 5–6) |
|
| Graphs of quadratic functions (Algebra 1 and Algebra 2) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to complete the square (vertex p, q)
| x² coefficient a | 2 |
| x coefficient b | -8 |
| Constant c | 3 |
| Vertex x-coordinate p | =-B2/(2*B1) |
| Vertex y-coordinate q | =B3-B2^2/(4*B1) |
Table to find the x-coordinate of the vertex (axis of symmetry)
| x² coefficient a | 1 |
| x coefficient b | -6 |
| Vertex x-coordinate p = −b/(2a) | =-B2/(2*B1) |
Table to find the y-coordinate of the vertex (max or min)
| x² coefficient a | 1 |
| x coefficient b | -6 |
| Constant c | 5 |
| Vertex y-coordinate q = c − b²/(4a) | =B3-B2^2/(4*B1) |
Table to check that vertex form matches the original
| x² coefficient a | 2 |
| x coefficient b | -8 |
| Constant c | 3 |
| x value to check | 1 |
| Vertex x-coordinate p | =-B2/(2*B1) |
| Vertex y-coordinate q | =B3-B2^2/(4*B1) |
| Value of ax²+bx+c | =B1*B4^2+B2*B4+B3 |
| Value of a(x−p)²+q | =B1*(B4-B5)^2+B6 |
After pasting, the upper rows (coefficients) are your inputs and the lower rows are calculated automatically.
"^" is a power, "*" is multiplication and "/" is division.
The first table is the example y = 2x² − 8x + 3. p is 2 and q is −5 (so y = 2(x − 2)² − 5).
The second and third tables are the example y = x² − 6x + 5. p is 3 and q is −4.
The fourth table is a check. For the same x (1 in the example), "value of ax²+bx+c" and "value of a(x−p)²+q" are both −3, so the two forms match.
"^" is a power, "*" is multiplication and "/" is division.
The first table is the example y = 2x² − 8x + 3. p is 2 and q is −5 (so y = 2(x − 2)² − 5).
The second and third tables are the example y = x² − 6x + 5. p is 3 and q is −4.
The fourth table is a check. For the same x (1 in the example), "value of ax²+bx+c" and "value of a(x−p)²+q" are both −3, so the two forms match.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to complete the square (vertex p, q)
| x² coefficient a | 2 |
| x coefficient b | -8 |
| Constant c | 3 |
| Vertex x-coordinate p | =-B2/(2*B1) |
| Vertex y-coordinate q | =B3-B2^2/(4*B1) |
Table to find the x-coordinate of the vertex (axis of symmetry)
| x² coefficient a | 1 |
| x coefficient b | -6 |
| Vertex x-coordinate p = −b/(2a) | =-B2/(2*B1) |
Table to find the y-coordinate of the vertex (max or min)
| x² coefficient a | 1 |
| x coefficient b | -6 |
| Constant c | 5 |
| Vertex y-coordinate q = c − b²/(4a) | =B3-B2^2/(4*B1) |
Table to check that vertex form matches the original
| x² coefficient a | 2 |
| x coefficient b | -8 |
| Constant c | 3 |
| x value to check | 1 |
| Vertex x-coordinate p | =-B2/(2*B1) |
| Vertex y-coordinate q | =B3-B2^2/(4*B1) |
| Value of ax²+bx+c | =B1*B4^2+B2*B4+B3 |
| Value of a(x−p)²+q | =B1*(B4-B5)^2+B6 |
The same formulas as in Excel work as is. Copy the whole table, paste it into cell A1, and replace the coefficients with your own numbers.
How to calculate it in Python
from fractions import Fraction
# Coefficients of the quadratic function y = ax² + bx + c (a fraction such as 3/4 can be written as Fraction(3, 4))
coef_a = Fraction(2) # x² coefficient (not 0)
coef_b = Fraction(-8) # x coefficient
coef_c = Fraction(3) # constant
vertex_x = -coef_b / (2 * coef_a) # vertex x-coordinate p = −b/(2a)
vertex_y = coef_c - coef_b ** 2 / (4 * coef_a) # vertex y-coordinate q = c − b²/(4a)
print(f"Vertex x-coordinate p: {vertex_x}")
print(f"Vertex y-coordinate q: {vertex_y}")
print(f"Vertex form: y = {coef_a} * (x - ({vertex_x}))**2 + ({vertex_y})")
if coef_a > 0:
print(f"Opens upward, so the minimum is {vertex_y} at x = {vertex_x}")
else:
print(f"Opens downward, so the maximum is {vertex_y} at x = {vertex_x}")
With the fractions module from the standard library, the calculation stays in exact fractions with no rounding error. This example completes the square on y = 2x² − 8x + 3. Running it shows p = 2 and q = −5 (so y = 2(x − 2)² − 5, with minimum −5). Change the coefficients and run it again.
How to write it in LaTeX and other math languages (copy and paste)
Completing the square (converting to vertex form)
y = a(x − p)² + q
y = a(x - p)^{2} + q
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>y</mi>
<mo>=</mo>
<mi>a</mi>
<msup>
<mrow><mo>(</mo><mi>x</mi><mo>−</mo><mi>p</mi><mo>)</mo></mrow>
<mn>2</mn>
</msup>
<mo>+</mo>
<mi>q</mi>
</mrow>
</math>
y = a(x - p)^2 + q
a*(x - p)^2 + q
y := a*(x - p)^2 + q;
y = a*(x - p)^2 + q;
y = a(x - p)^2 + q
\(x\)-coordinate of the vertex (the axis of symmetry)
p = −b/(2a)
p = -\frac{b}{2a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>p</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>b</mi>
<mrow><mn>2</mn><mi>a</mi></mrow>
</mfrac>
</mrow>
</math>
p = -b/(2a)
-b/(2*a)
p := -b/(2*a);
p = -b/(2*a);
p = −b/(2a)
\(y\)-coordinate of the vertex (maximum or minimum value)
q = c − b²/(4a)
q = c - \frac{b^{2}}{4a}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>q</mi>
<mo>=</mo>
<mi>c</mi>
<mo>−</mo>
<mfrac>
<msup><mi>b</mi><mn>2</mn></msup>
<mrow><mn>4</mn><mi>a</mi></mrow>
</mfrac>
</mrow>
</math>
q = c - b^2/(4a)
c - b^2/(4*a)
q := c - b^2/(4*a);
q = c - b^2/(4*a);
q = c − b^2/(4a)
How to have ChatGPT do the calculation
You are a math assistant for quadratic functions. Do the following calculation by actually running Python, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Complete the square on the quadratic function y = 2x² − 8x + 3. Show each of the following: 1. The steps of completing the square (factor out the x² coefficient → add and subtract the square of half the x coefficient → write the perfect square) 2. The vertex form y = a(x − p)² + q, the vertex (p, q) and the axis of symmetry (as reduced fractions) 3. Which way the graph opens (upward or downward), and the maximum or minimum value In Python, use the fractions module from the standard library to calculate exactly. Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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