Vertex Form Calculator With Steps

Standard form	$y = x^{2} - 6x + 5$
Factored form	$y = \left(x - 5\right)\left(x - 1\right)$
Vertex form	$y = \left(x - 3\right)^2 - 4$
Vertex	$\left(3,\ -4\right)$
Axis of symmetry	$x = 3$
y-intercept	$\left(0,\ 5\right)$
Discriminant	$\Delta = 16$

Standard form

y = x^{2} - 6x + 5

Factored form

y = \left(x - 5\right)\left(x - 1\right)

Vertex form

y = \left(x - 3\right)^2 - 4

Vertex

\left(3,\ -4\right)

Axis of symmetry

x = 3

y-intercept

\left(0,\ 5\right)

Discriminant

\Delta = 16

Graph of the parabola with its roots and vertex marked: sampled values across the visible range
x	y
-1.8	19.04
-0.84	10.746
0.12	4.2944
1.08	-0.3136
2.04	-3.0784
3	-4
3.96	-3.0784
4.92	-0.3136
5.88	4.2944
6.84	10.746
7.8	19.04

-1.8

19.04

-0.84

10.746

0.12

4.2944

1.08

-0.3136

2.04

-3.0784

-4

3.96

-3.0784

4.92

-0.3136

5.88

4.2944

6.84

10.746

7.8

19.04

Quadratic formula, step by step

1Identify the coefficients
$ax^2 + bx + c = 0$
$x^{2} - 6x + 5 = 0$
a = 1, b = -6, c = 5
2Discriminant
$\Delta = b^2 - 4ac$
$\Delta = \left(-6\right)^2 - 4(1)5 = 16$
Positive: two different real roots.
3Quadratic formula
$x = \frac{-b \pm \sqrt{\Delta}}{2a}$
$x = \frac{6 \pm \sqrt{16}}{2}$
4Square root of the discriminant
$\sqrt{16}$
$4$
16 is a perfect square.
5The two roots
$x = \frac{6 \pm 4}{2}$
$x_1 = 5, \quad x_2 = 1$

Steps

1Divide by a
$x^{2} - 6x + 5 = 0$
$x^2 - 6x = -5$
a is already 1, so nothing changes.
2Add the square of half the x coefficient
$\left(\frac{b}{2a}\right)^2$
$\left(-3\right)^2 = 9$
Add this to both sides.
3Write the left side as a square
$x^2 - 6x + 9 = 4$
$\left(x - 3\right)^2 = 4$
The right side equals the discriminant over 4a squared: 16 / 4.
4Take the square root of both sides
$x - 3 = \pm\sqrt{4}$
$x - 3 = \pm 2$
5Vertex form
$y = x^{2} - 6x + 5$
$y = \left(x - 3\right)^2 - 4$
The same completing-the-square work, kept as an expression, gives the vertex.

About Vertex Form Calculator

This vertex form calculator rewrites y = ax² + bx + c as y = a(x - h)² + k, so the highest or lowest point of the parabola is visible straight away. It is free, needs no sign-up and runs in your browser, so nothing you enter is uploaded.

The vertex has x coordinate h = -b/(2a), and k is the value of the quadratic at h. The line x = h is the axis of symmetry. If a is positive the parabola opens up and k is the minimum value; if a is negative it opens down and k is the maximum.

The page opens on x² - 6x + 5, whose vertex is (3, -4). The completing-the-square steps are shown as well, since that is how vertex form is usually derived in class.

How to use Vertex Form Calculator

Choose what to show: Roots, Discriminant, Vertex or Completing the square.
Enter a, b and c in their boxes, or switch to Equation and type something like 2x^2 - 7x + 3 = 0.
Read the headline answer. It updates as you type, with no calculate button.
Open the steps underneath to follow the substitution into the quadratic formula, or the completing-the-square route.
Use the table for the factored form, vertex form, axis of symmetry and y-intercept, and the graph to see where the parabola crosses the x axis.

Features

Exact roots: rational, simplified radical or complex, never only a rounded decimal.
Discriminant with a reading: two real roots, one repeated root or a complex pair.
Vertex, axis of symmetry, y-intercept and whether the parabola opens up or down.
Factored form over the rationals when it exists, with the complex factorisation otherwise.
Completing the square written out, ending in vertex form.
Quadratic formula steps in KaTeX, with a copy button for the LaTeX.
Interactive plot with roots and vertex marked, zoom, pan and PNG export.
Accepts integers, decimals, fractions and a full equation.

Vertex form from standard form

Use the vertex formula, or complete the square. Both give the same h and k.

h = -b / (2a)
k = c - b² / (4a)
y = a(x - h)² + k

Worked examples

y = x² - 6x + 5 gives y = (x - 3)² - 4, vertex (3, -4)
h = 6/2 = 3 and k = 5 - 9 = -4. The roots are 1 and 5, symmetric about x = 3.
y = -2x² + 4x + 1 gives y = -2(x - 1)² + 3, vertex (1, 3)
a is negative, so the parabola opens downward and 3 is the maximum value.

Privacy: how it works

Everything is calculated by code running in your browser tab. The coefficients you enter are not sent to Ulkaa or to any other server, and there is nothing to sign in to.

Tips

Check an answer by substituting a root back into the equation: the left side should come out as 0.
If the roots look awkward, look at the discriminant first. A perfect square means rational roots and a clean factorisation.
The sum of the roots is -b/a and their product is c/a, which is a quick check on any pair of roots.

Frequently asked questions

Is it really free?

Yes. There is no price, no trial and no credit card. Ulkaa Tool does the work with your own device, so there is no server bill to pass on to you.

Do I need an account?

No. There is no sign-up and no login. Open the page and use it. An Ulkaa account is not needed for anything on this tool.

Does it run in my browser?

Yes. Coefficients are processed by code running in your browser tab, on your own device.

Is my data uploaded?

No. Coefficients are never sent to Ulkaa or to anyone else. You can check this yourself by watching the Network tab in your browser developer tools while you use the tool.

How do I convert standard form to vertex form?

Find h = -b/(2a), then k = c - b²/(4a), and write y = a(x - h)² + k. Completing the square gives the same result.

What does the sign of a tell me?

A positive a means the parabola opens upward and the vertex is the lowest point. A negative a means it opens downward and the vertex is the highest point.

Standard form	$y = x^{2} - 6x + 5$
Factored form	$y = \left(x - 5\right)\left(x - 1\right)$
Vertex form	$y = \left(x - 3\right)^2 - 4$
Vertex	$\left(3,\ -4\right)$
Axis of symmetry	$x = 3$
y-intercept	$\left(0,\ 5\right)$
Discriminant	$\Delta = 16$

Standard form

y = x^{2} - 6x + 5

Factored form

y = \left(x - 5\right)\left(x - 1\right)

Vertex form

y = \left(x - 3\right)^2 - 4

Vertex

\left(3,\ -4\right)

Axis of symmetry

x = 3

y-intercept

\left(0,\ 5\right)

Discriminant

\Delta = 16

Graph of the parabola with its roots and vertex marked: sampled values across the visible range
x	y
-1.8	19.04
-0.84	10.746
0.12	4.2944
1.08	-0.3136
2.04	-3.0784
3	-4
3.96	-3.0784
4.92	-0.3136
5.88	4.2944
6.84	10.746
7.8	19.04

-1.8

19.04

-0.84

10.746

0.12

4.2944

1.08

-0.3136

2.04

-3.0784

-4

3.96

-3.0784

4.92

-0.3136

5.88

4.2944

6.84

10.746

7.8

19.04

Quadratic formula, step by step

1Identify the coefficients
$ax^2 + bx + c = 0$
$x^{2} - 6x + 5 = 0$
a = 1, b = -6, c = 5
2Discriminant
$\Delta = b^2 - 4ac$
$\Delta = \left(-6\right)^2 - 4(1)5 = 16$
Positive: two different real roots.
3Quadratic formula
$x = \frac{-b \pm \sqrt{\Delta}}{2a}$
$x = \frac{6 \pm \sqrt{16}}{2}$
4Square root of the discriminant
$\sqrt{16}$
$4$
16 is a perfect square.
5The two roots
$x = \frac{6 \pm 4}{2}$
$x_1 = 5, \quad x_2 = 1$

Steps

1Divide by a
$x^{2} - 6x + 5 = 0$
$x^2 - 6x = -5$
a is already 1, so nothing changes.
2Add the square of half the x coefficient
$\left(\frac{b}{2a}\right)^2$
$\left(-3\right)^2 = 9$
Add this to both sides.
3Write the left side as a square
$x^2 - 6x + 9 = 4$
$\left(x - 3\right)^2 = 4$
The right side equals the discriminant over 4a squared: 16 / 4.
4Take the square root of both sides
$x - 3 = \pm\sqrt{4}$
$x - 3 = \pm 2$
5Vertex form
$y = x^{2} - 6x + 5$
$y = \left(x - 3\right)^2 - 4$
The same completing-the-square work, kept as an expression, gives the vertex.

About Vertex Form Calculator

The page opens on x² - 6x + 5, whose vertex is (3, -4). The completing-the-square steps are shown as well, since that is how vertex form is usually derived in class.

Vertex Form Calculator

Vertex

Forms of the same parabola

Parabola

Quadratic formula, step by step

Steps

About Vertex Form Calculator

How to use Vertex Form Calculator

Features

Vertex form from standard form

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

How do I convert standard form to vertex form?

What does the sign of a tell me?

Vertex Form Calculator

Vertex

Forms of the same parabola

Parabola

Quadratic formula, step by step

Steps

About Vertex Form Calculator

How to use Vertex Form Calculator

Features

Vertex form from standard form

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

How do I convert standard form to vertex form?

What does the sign of a tell me?

Vertex Form Calculator

Vertex

Forms of the same parabola

Parabola

Quadratic formula, step by step

Steps

About Vertex Form Calculator

How to use Vertex Form Calculator

Features

Vertex form from standard form

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

How do I convert standard form to vertex form?

What does the sign of a tell me?

More Quadratic Formula Calculator pages

Vertex Form Calculator

Vertex

Forms of the same parabola

Parabola

Quadratic formula, step by step

Steps

About Vertex Form Calculator

How to use Vertex Form Calculator

Features

Vertex form from standard form

Worked examples

Privacy: how it works

Tips

Frequently asked questions

Is it really free?

Do I need an account?

Does it run in my browser?

Is my data uploaded?

How do I convert standard form to vertex form?

What does the sign of a tell me?

More Quadratic Formula Calculator pages