Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola and x = h is its axis of symmetry. The a is the same number as in standard form: above zero the curve opens up, below zero it opens down. From standard form, h = -b/(2a) and k = c - b²/(4a).

How do you complete the square?

Take 2x² + 8x + 5. Take the 2 out of the x terms to get 2(x² + 4x) + 5. Half of 4 is 2 and its square is 4, so add 4 inside the bracket and take 2 × 4 = 8 back off outside: 2(x² + 4x + 4) - 8 + 5. The bracket is (x + 2)², so the vertex form is 2(x + 2)² - 3 and the vertex is (-2, -3). The working writes these three lines out for whatever a, b and c you enter.

Can it graph the parabola?

Yes. The curve is drawn beside the answer with the vertex marked and any real roots shown on the x axis, and the working names the intercepts: where the curve crosses the x axis, and the y-intercept (0, c).

Can it convert vertex form to standard or factored form?

Yes. Start from vertex form, under the working, takes a, h and k and multiplies them out with b = -2ah and c = ah² + k. Solve this one then loads the result, and the working gives its roots and, when they are real, its factored form.

Vertex Form Calculator

Enter a, b and c from standard form and the page rewrites the quadratic as a(x - h)² + k. The working completes the square with your numbers, then gives the vertex (h, k), the axis of symmetry and which way the parabola opens, and the curve is drawn with its vertex and any x-intercepts marked. Whole-number coefficients keep exact fractions, so x² + x + 1 becomes (x + 1/2)² + 3/4. To go the other way, from a vertex form back to standard form, use Start from vertex form under the working.

Vertex form(x + 1/2)² - 25/4a=1 b=1 c=-6 discriminant 25

A parabola opening upward, vertex at (-0.5, -6.25). It crosses the x axis at 2 and -3.

a
b
c
Give me

Vertex form: (x + 1/2)² - 25/4
h: -1/2
k: -25/4
Opens: Up

The same equation rewritten as a(x - h)² + k, with the vertex, the axis of symmetry and the curve drawn.

The working

x = (-b ± √(b² - 4ac)) / (2a), with a = 1, b = 1, c = -6
D = b² - 4ac = (1)² - 4(1)(-6) = 25
√D = 5, so x = 2, -3
Both roots come from q = -(b + sign(b)√D)/2, read as q/a and c/q. That is the stable form: written as (-b ± √D)/(2a), the smaller root loses digits whenever b² is far larger than 4ac.
h = -b/(2a) = -1/2 = -1/2
k = c - b²/(4a) = -6 - 1/4 = -25/4
Vertex form: (x + 1/2)² - 25/4, vertex (-1/2, -25/4), axis of symmetry x = -1/2
Completing the square: x² + x - 6 = (x² + x) - 6
Half of 1 is 1/2, and its square is 1/4. Add 1/4 inside the bracket and take 1/4 back off outside: (x² + x + 1/4) - 1/4 - 6
The bracket is the perfect square (x + 1/2)², so x² + x - 6 = (x + 1/2)² - 25/4
Factored form: x² + x - 6 = (x - 2)(x + 3)
Intercepts: the curve crosses the x axis at (2, 0) and (-3, 0), and the y axis at (0, -6).
Exact: a, b and c are whole numbers, so the discriminant is exact and the roots are given as fractions or surds beside their decimals.

Each root, put back into the equation

A root is only a root if it makes the left-hand side zero. The residual is the left-hand side worked out at the root this page computed, so one marked as rounding is the last-digit error of double arithmetic and not a second answer.

x₁ = 2: 1(2)² + 1(2) - 6 = 0
x₂ = -3: 1(-3)² + 1(-3) - 6 = 0

Start from vertex form

Type a, h and k from y = a(x - h)² + k to multiply it out into standard form y = ax² + bx + c. Solve this one sends the result to the solver for its roots, factors and graph.

a
h
k

y = 2x² - 12x + 10

Start from y = 2(x - 3)² - 8, so a = 2, h = 3 and k = -8.
Square the bracket: (x - h)² = x² - 2hx + h², so a(x - h)² + k = ax² - 2ahx + ah² + k.
b = -2ah = -2(2)(3) = -12
c = ah² + k = 2(3)² - 8 = 10
Standard form: y = 2x² - 12x + 10

Three equations worth trying

One of each kind the discriminant allows. Loading one keeps the mode you are in.

a=1 b=1 c=-6D = 25
y = (x + 1/2)² - 25/4
a=1 b=2 c=1D = 0
y = (x + 1)²
a=1 b=0 c=1D = -4
y = x² + 1

The discriminant decides how many roots there are. b² - 4ac above zero gives two real roots, exactly zero gives one repeated root, and below zero gives a conjugate pair with an imaginary part. That is why it sits in the readout next to the answer rather than buried in the working: it is the number that says which of the three answers you are looking at. With a at zero there is no x² term at all, so the equation is linear and is solved as one.

Common questions

What is vertex form?: Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola and x = h is its axis of symmetry. The a is the same number as in standard form: above zero the curve opens up, below zero it opens down. From standard form, h = -b/(2a) and k = c - b²/(4a).
How do you complete the square?: Take 2x² + 8x + 5. Take the 2 out of the x terms to get 2(x² + 4x) + 5. Half of 4 is 2 and its square is 4, so add 4 inside the bracket and take 2 × 4 = 8 back off outside: 2(x² + 4x + 4) - 8 + 5. The bracket is (x + 2)², so the vertex form is 2(x + 2)² - 3 and the vertex is (-2, -3). The working writes these three lines out for whatever a, b and c you enter.
Can it graph the parabola?: Yes. The curve is drawn beside the answer with the vertex marked and any real roots shown on the x axis, and the working names the intercepts: where the curve crosses the x axis, and the y-intercept (0, c).
Can it convert vertex form to standard or factored form?: Yes. Start from vertex form, under the working, takes a, h and k and multiplies them out with b = -2ah and c = ah² + k. Solve this one then loads the result, and the working gives its roots and, when they are real, its factored form.

Every answer shows its discriminant, and each root is substituted back so you can see the residual. Whole-number coefficients up to 10 million give exact results: rational roots as fractions, irrational and complex roots in surd form, each with a decimal beside it; other coefficients are solved in double precision, where the last digits can round. With a at zero this is not a quadratic: it is solved as a linear equation, and vertex form is declined.