Quadratic formula practice worksheet with eight equations to solve.

Here are the step-by-step solutions for each equation using the quadratic formula.

The quadratic formula is:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

For every problem, we first identify $a$, $b$, and $c$ from the standard form $ax^2 + bx + c = 0$. Then we plug them into the formula.

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1) $m^2 - 5m - 14 = 0$

* Identify coefficients: $a = 1$, $b = -5$, $c = -14$
* Plug into formula:
$$m = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(-14)}}{2(1)}$$
$$m = \frac{5 \pm \sqrt{25 + 56}}{2}$$
$$m = \frac{5 \pm \sqrt{81}}{2}$$
$$m = \frac{5 \pm 9}{2}$$
* Solve for both cases:
Case 1: $m = \frac{5 + 9}{2} = \frac{14}{2} = 7$
Case 2: $m = \frac{5 - 9}{2} = \frac{-4}{2} = -2$

2) $b^2 - 4b + 4 = 0$

* Identify coefficients: $a = 1$, $b = -4$, $c = 4$
* Plug into formula:
$$b = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(4)}}{2(1)}$$
$$b = \frac{4 \pm \sqrt{16 - 16}}{2}$$
$$b = \frac{4 \pm \sqrt{0}}{2}$$
$$b = \frac{4}{2} = 2$

3) $2m^2 + 2m - 12 = 0$

* Identify coefficients: $a = 2$, $b = 2$, $c = -12$
* Plug into formula:
$$m = \frac{-2 \pm \sqrt{2^2 - 4(2)(-12)}}{2(2)}$$
$$m = \frac{-2 \pm \sqrt{4 + 96}}{4}$$
$$m = \frac{-2 \pm \sqrt{100}}{4}$$
$$m = \frac{-2 \pm 10}{4}$$
* Solve for both cases:
Case 1: $m = \frac{-2 + 10}{4} = \frac{8}{4} = 2$
Case 2: $m = \frac{-2 - 10}{4} = \frac{-12}{4} = -3$

4) $2x^2 - 3x - 5 = 0$

* Identify coefficients: $a = 2$, $b = -3$, $c = -5$
* Plug into formula:
$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(-5)}}{2(2)}$$
$$x = \frac{3 \pm \sqrt{9 + 40}}{4}$$
$$x = \frac{3 \pm \sqrt{49}}{4}$$
$$x = \frac{3 \pm 7}{4}$$
* Solve for both cases:
Case 1: $x = \frac{3 + 7}{4} = \frac{10}{4} = \frac{5}{2}$ (or 2.5)
Case 2: $x = \frac{3 - 7}{4} = \frac{-4}{4} = -1$

5) $x^2 + 4x + 3 = 0$

* Identify coefficients: $a = 1$, $b = 4$, $c = 3$
* Plug into formula:
$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(3)}}{2(1)}$$
$$x = \frac{-4 \pm \sqrt{16 - 12}}{2}$$
$$x = \frac{-4 \pm \sqrt{4}}{2}$$
$$x = \frac{-4 \pm 2}{2}$$
* Solve for both cases:
Case 1: $x = \frac{-4 + 2}{2} = \frac{-2