Completing the square questions and answers

Worked solution

1. Half of 6 is 3: $(x + 3)^2 = x^2 + 6x + 9$.
2. $x^2 + 6x + 4 = (x + 3)^2 - 9 + 4$
3. $= (x + 3)^2 - 5$

Worked solution

1. $(x - 5)^2 = x^2 - 10x + 25$
2. $x^2 - 10x + 30 = (x - 5)^2 - 25 + 30 = (x - 5)^2 + 5$

Worked solution

1. $\left(x + \tfrac{3}{2}\right)^2 = x^2 + 3x + \tfrac{9}{4}$
2. $x^2 + 3x + 1 = \left(x + \tfrac{3}{2}\right)^2 - \tfrac{9}{4} + 1 = \left(x + \tfrac{3}{2}\right)^2 - \tfrac{5}{4}$

Worked solution

1. $x^2 + 8x + 11 = (x + 4)^2 - 16 + 11 = (x + 4)^2 - 5$
2. The minimum is when $x + 4 = 0$, so $x = -4$ and $y = -5$.
3. Turning point: $(-4, -5)$

Worked solution

1. $2x^2 + 12x + 5 = 2(x^2 + 6x) + 5$
2. $x^2 + 6x = (x + 3)^2 - 9$
3. $2\left((x + 3)^2 - 9\right) + 5 = 2(x + 3)^2 - 18 + 5 = 2(x + 3)^2 - 13$

Worked solution

1. $3x^2 - 12x + 7 = 3(x^2 - 4x) + 7$
2. $x^2 - 4x = (x - 2)^2 - 4$
3. $3\left((x - 2)^2 - 4\right) + 7 = 3(x - 2)^2 - 12 + 7 = 3(x - 2)^2 - 5$

Worked solution

1. $x^2 + 6x - 3 = (x + 3)^2 - 9 - 3 = (x + 3)^2 - 12$
2. $(x + 3)^2 = 12$, so $x + 3 = \pm\sqrt{12} = \pm 2\sqrt{3}$
3. $x = -3 \pm 2\sqrt{3}$

Worked solution

1. $(x - 4)^2 - 3 = x^2 - 8x + 16 - 3 = x^2 - 8x + 13$
2. Comparing: $p = -8$ and $q = 13$.

Worked solution

1. $x^2 - 4x + 7 = (x - 2)^2 - 4 + 7 = (x - 2)^2 + 3$
2. $(x - 2)^2 \ge 0$ for every $x$, because a square is never negative.
3. So $(x - 2)^2 + 3 \ge 3 > 0$: the expression is always positive.

Worked solution

1. $x^2 - 5x + 2 = \left(x - \tfrac{5}{2}\right)^2 - \tfrac{25}{4} + 2 = \left(x - \tfrac{5}{2}\right)^2 - \tfrac{17}{4}$
2. $\left(x - \tfrac{5}{2}\right)^2 = \tfrac{17}{4}$, so $x - \tfrac{5}{2} = \pm\tfrac{\sqrt{17}}{2}$
3. $x = \dfrac{5 \pm \sqrt{17}}{2}$