The Quadratic Formula: Practice

Read off $a = 2$, $b = -4$, $c = 3$, so $-b = 4$ and $D = 16 - 24 = -8$, with $\sqrt{-8} = 2i\sqrt{2}$.

$x = \frac{4 \pm 2i\sqrt{2}}{4} = \frac{2 \pm i\sqrt{2}}{2}$

Reduce numerator and denominator by the common factor $2$.

Read off $a = 2$, $b = 2$, $c = 5$, so $D = 4 - 40 = -36$, with $\sqrt{-36} = 6i$.

$x = \frac{-2 \pm 6i}{4} = \frac{2(-1 \pm 3i)}{4} = \frac{-1 \pm 3i}{2}$

Reduce by the common factor $2$.

With $b = -4$, the numerator uses $-b = -(-4) = 4$, so the roots center on $+2$.

$x = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}$

The student kept $-4$ in the numerator, flipping the sign to the wrong $-2 \pm \sqrt{3}$.

First reach standard form by moving every term left: $x^2 - 6x + 13 = 0$. Then $-b = 6$ and $D = 36 - 52 = -16$, with $\sqrt{-16} = 4i$.

$x = \frac{6 \pm 4i}{2} = 3 \pm 2i$

A repeated root needs $D = k^2 - 4(1)(9) = 0$.

$k^2 - 36 = 0 \quad\Rightarrow\quad k^2 = 36 \quad\Rightarrow\quad k = \pm 6$

Then $x^2 + 6x + 9 = (x + 3)^2$ and $x^2 - 6x + 9 = (x - 3)^2$.

Read off $a = 5$, $b = -2$, $c = 1$, so $-b = 2$ and $D = 4 - 20 = -16$, with $\sqrt{-16} = 4i$.

$x = \frac{2 \pm 4i}{10} = \frac{2(1 \pm 2i)}{10} = \frac{1 \pm 2i}{5}$

Reduce by the common factor $2$.

Read off $a = 4$, $b = 4$, $c = 1$, and the discriminant is $16 - 16 = 0$.

$x = \frac{-4 \pm \sqrt{0}}{8} = \frac{-4}{8} = -\tfrac{1}{2}$

A repeated root, since $4x^2 + 4x + 1 = (2x + 1)^2$.

Completing the square needs a leading coefficient of $1$, so divide every term by $a$, which is allowed because $a \ne 0$.

$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$

From here you move the constant across, add $\left(\frac{b}{2a}\right)^2$ to both sides, and finish.

Read off $a = 1$, $b = -10$, $c = 18$, so $-b = 10$ and $D = 100 - 72 = 28$, with $\sqrt{28} = 2\sqrt{7}$.

$x = \frac{10 \pm 2\sqrt{7}}{2} = 5 \pm \sqrt{7}$

Read off $a = 3$, $b = 6$, $c = 2$, so $D = 36 - 24 = 12$, with $\sqrt{12} = 2\sqrt{3}$.

$x = \frac{-6 \pm 2\sqrt{3}}{6} = \frac{2(-3 \pm \sqrt{3})}{6} = \frac{-3 \pm \sqrt{3}}{3}$

Reduce by the common factor $2$.

Read off $a = 1$, $b = -8$, $c = 5$, so $-b = 8$ and $D = 64 - 20 = 44$, with $\sqrt{44} = 2\sqrt{11}$.

$x = \frac{8 \pm 2\sqrt{11}}{2} = 4 \pm \sqrt{11}$

The discriminant is right: $(-3)^2 - 4(2)(-2) = 9 + 16 = 25$. But the denominator must be $2a = 2(2) = 4$.

$x = \frac{3 \pm \sqrt{25}}{4} = \frac{3 \pm 5}{4}$

So $x = 2$ or $x = -\tfrac{1}{2}$. The student divided by $2$ instead of $4$.