The Quadratic Formula: Practice

Read off $a = 1$, $b = 6$, $c = 2$, so $D = 36 - 8 = 28$, with $\sqrt{28} = 2\sqrt{7}$.

$x = \frac{-6 \pm 2\sqrt{7}}{2} = -3 \pm \sqrt{7}$

Both parts of the numerator are divided by $2$.

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

$x = \frac{2 \pm 4i}{2} = 1 \pm 2i$

Divide both the $2$ and the $4i$ by $2$.

Read off $a = 2$, $b = 1$, $c = -6$, so $-4ac = +48$ and $D = 1 + 48 = 49$.

$x = \frac{-1 \pm \sqrt{49}}{4} = \frac{-1 \pm 7}{4}$

So $x = \frac{6}{4} = \tfrac{3}{2}$ or $x = \frac{-8}{4} = -2$.

Move every term to the left side so the other side is $0$.

$2x^2 = 5x - 1 \quad\Rightarrow\quad 2x^2 - 5x + 1 = 0$

So $a = 2$, $b = -5$, $c = 1$. Subtracting $5x$ makes $b$ negative, and moving $-1$ makes $c = +1$.

Read off $a = 1$, $b = -6$, $c = 7$, so $-b = 6$ and $D = 36 - 28 = 8$, with $\sqrt{8} = 2\sqrt{2}$.

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

Read off $a = 1$, $b = 2$, $c = 2$, so $D = 4 - 8 = -4$, with $\sqrt{-4} = 2i$.

$x = \frac{-2 \pm 2i}{2} = -1 \pm i$

Divide both parts of the numerator by $2$.

Factor out the largest perfect square, $48 = 16 \times 3$.

$\sqrt{48} = \sqrt{16}\,\sqrt{3} = 4\sqrt{3}$

Since $3$ has no square factor left, $4\sqrt{3}$ is fully simplified.

Read off $a = 1$, $b = -3$, $c = -10$, so $-b = 3$ and $-4ac = +40$, giving $D = 9 + 40 = 49$.

$x = \frac{3 \pm \sqrt{49}}{2} = \frac{3 \pm 7}{2}$

So $x = 5$ or $x = -2$.

Compute the discriminant with $a = 1$, $b = 1$, $c = 4$.

$D = 1^2 - 4(1)(4) = 1 - 16 = -15$

A negative discriminant means no real solutions; both roots are complex.

Read off $a = 9$, $b = 0$, $c = -4$, so the $-b$ term is $0$.

$x = \frac{0 \pm \sqrt{-4(9)(-4)}}{18} = \frac{\pm\sqrt{144}}{18} = \frac{\pm 12}{18} = \pm\tfrac{2}{3}$

Read off $a = 1$, $b = -2$, $c = 10$, so $-b = 2$ and $D = 4 - 40 = -36$, with $\sqrt{-36} = 6i$.

$x = \frac{2 \pm 6i}{2} = 1 \pm 3i$

With $a = 2$, $b = 7$, $c = -4$, and $c$ negative so $-4ac$ is positive.

$D = 7^2 - 4(2)(-4) = 49 + 32 = 81$

A perfect square, so the equation has two rational roots.