Completing the Square and Vertex Form, Step by Step

Completing the square rewrites $ax^2 + bx + c$ as $a(x - h)^2 + k$. In this vertex form you can read the turning point $(h, k)$ straight off the equation. The key move is to add and subtract the square of half the $x$-coefficient.

Why vertex form is useful

• The vertex (turning point) is $(h, k)$.
• The axis of symmetry is $x = h$.
• If $a > 0$ the minimum value is $k$; if $a < 0$ the maximum is $k$.

Worked example 1: $x^2 + 6x + 5$

1. Half of 6 is 3, and $3^2 = 9$.
2. Add and subtract 9: $x^2 + 6x + 9 - 9 + 5$.
3. The first three terms make a square: $(x + 3)^2 - 4$.

So $y = (x + 3)^2 - 4$ with vertex $(-3, -4)$.

Worked example 2: when $a \ne 1$

$y = 2x^2 - 8x + 3$

1. Factor 2 out of the $x$ terms: $y = 2(x^2 - 4x) + 3$.
2. Half of $-4$ is $-2$; $(-2)^2 = 4$: $y = 2[(x - 2)^2 - 4] + 3$.
3. Multiply out the bracket: $y = 2(x - 2)^2 - 8 + 3 = 2(x - 2)^2 - 5$.

Vertex $(2, -5)$, and since $a = 2 > 0$ it is a minimum.

Solving equations by completing the square

Set the vertex form equal to zero: $(x + 3)^2 - 4 = 0$ gives $(x + 3)^2 = 4$, so $x + 3 = \pm 2$ and $x = -1$ or $x = -5$.

Common mistakes

• Forgetting to subtract the number you added.
• Not multiplying the subtracted square by $a$ when $a \ne 1$.
• Getting the sign of $h$ wrong: $(x + 3)^2$ means $h = -3$.

Practice questions

1. Write $x^2 - 4x + 1$ in vertex form.
2. Find the vertex of $y = x^2 + 2x - 8$.
3. Write $-x^2 + 4x - 1$ in vertex form.

Answers: 1) $(x - 2)^2 - 3$ 2) $(-1, -9)$ 3) $-(x - 2)^2 + 3$

The vertex form calculator shows every step and graphs the parabola. To find roots, use the quadratic equation solver, and read three ways to solve a quadratic.

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