Completing the square

Factorising review, perfect squares, completing the square for x² + bx, writing quadratics in the form (x + h)² + k, and solving by completing the square.

The method

Example 1●●

Write $x^2 - 8x + 3$ in the form $(x + h)^2 + k$.

Half of $-8$ is $-4$; $(-4)^2 = 16$.

$x^2 - 8x + 16 - 16 + 3 = (x - 4)^2 - 13$

Solving

Example 2●●●

Solve $x^2 + 6x + 2 = 0$ by completing the square.

$x^2 + 6x + 9 = -2 + 9$, so $(x + 3)^2 = 7$.

$x + 3 = \pm\sqrt{7}$, so $x = -3 \pm \sqrt{7}$.

Flashcards

Complete the square: $x^2 + 10x$

Add $25$: $(x + 5)^2$

$x^2 + 6x + 5$

$(x + 3)^2 - 4$

$(x - 2)^2 = 5$

$x = 2 \pm \sqrt{5}$

Practice

Write $x^2 - 8x + 3$ as $(x + h)^2 + k$.
Solve $x^2 + 6x + 2 = 0$.
What number completes the square for $x^2 + 10x$?
$x^2 + 6x + 5 = (x + 3)^2 + k$. Find $k$.
Solve $x^2 - 4x - 1 = 0$, to 2 d.p. Enter both.
What number completes the square for $x^2 + 3x$? Give a fraction.
Solve $(x + 3)^2 = 16$.
$x^2 - 10x + 23 = (x - h)^2 + k$. Enter $h$, then $k$.