Quadratic Formula Worksheets - Math Monks - Free Printable

Let's solve each quadratic equation using the quadratic formula:

The quadratic formula is:

$$
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
$$

For a quadratic equation in the form:
$$
ax^2 + bx + c = 0
$$

We'll go through each problem one by one.

---

1) $-y^2 - 2y + 19 = -29$

First, move all terms to one side:

$$
-y^2 - 2y + 19 + 29 = 0 \Rightarrow -y^2 - 2y + 48 = 0
$$

Multiply both sides by $-1$ to make the leading coefficient positive:

$$
y^2 + 2y - 48 = 0
$$

Now apply the quadratic formula:
- $a = 1$, $b = 2$, $c = -48$

$$
y = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-48)}}{2(1)} = \frac{-2 \pm \sqrt{4 + 192}}{2} = \frac{-2 \pm \sqrt{196}}{2}
$$

$$
\sqrt{196} = 14 \Rightarrow y = \frac{-2 \pm 14}{2}
$$

So:
- $y = \frac{-2 + 14}{2} = \frac{12}{2} = 6$
- $y = \frac{-2 - 14}{2} = \frac{-16}{2} = -8$

✔ Solutions: $y = 6, -8$

---

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

Already in standard form.
- $a = 3$, $b = -1$, $c = -2$

$$
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-2)}}{2(3)} = \frac{1 \pm \sqrt{1 + 24}}{6} = \frac{1 \pm \sqrt{25}}{6}
$$

$$
\sqrt{25} = 5 \Rightarrow x = \frac{1 \pm 5}{6}
$$

- $x = \frac{1 + 5}{6} = \frac{6}{6} = 1$
- $x = \frac{1 - 5}{6} = \frac{-4}{6} = -\frac{2}{3}$

✔ Solutions: $x = 1, -\frac{2}{3}$

---

3) $-2x^2 + 17x - 12 = 9$

Move 9 to the left:

$$
-2x^2 + 17x - 12 - 9 = 0 \Rightarrow -2x^2 + 17x - 21 = 0
$$

Multiply by $-1$:

$$
2x^2 - 17x + 21 = 0
$$

- $a = 2$, $b = -17$, $c = 21$

$$
x = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(2)(21)}}{2(2)} = \frac{17 \pm \sqrt{289 - 168}}{4} = \frac{17 \pm \sqrt{121}}{4}
$$

$$
\sqrt{121} = 11 \Rightarrow x = \frac{17 \pm 11}{4}
$$

- $x = \frac{17 + 11}{4} = \frac{28}{4} = 7$
- $x = \frac{17 - 11}{4} = \frac{6}{4} = \frac{3}{2}$

✔ Solutions: $x = 7, \frac{3}{2}$

---

4) $2x^2 - 6x - 13 = 43$

Move 43 to the left:

$$
2x^2 - 6x - 13 - 43 = 0 \Rightarrow 2x^2 - 6x - 56 = 0
$$

Divide entire equation by 2 to simplify:

$$
x^2 - 3x - 28 = 0
$$

Now use quadratic formula:
- $a = 1$, $b = -3$, $c = -28$

$$
x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-28)}}{2(1)} = \frac{3 \pm \sqrt{9 + 112}}{2} = \frac{3 \pm \sqrt{121}}{2}
$$

$$
\sqrt{121} = 11 \Rightarrow x = \frac{3 \pm 11}{2}
$$

- $x = \frac{3 + 11}{2} = \frac{14}{2} = 7$
- $x = \frac{3 - 11}{2} = \frac{-8}{2} = -4$

✔ Solutions: $x = 7, -4$

---

5) $4x^2 - 28x - 41 = 0$

Already in standard form.
- $a = 4$, $b = -28$, $c = -41$

$$
x = \frac{-(-28) \pm \sqrt{(-28)^2 - 4(4)(-41)}}{2(4)} = \frac{28 \pm \sqrt{784 + 656}}{8} = \frac{28 \pm \sqrt{1440}}{8}
$$

Simplify $\sqrt{1440}$:

$$
\sqrt{1440} = \sqrt{144 \cdot 10} = 12\sqrt{10}
$$

$$
x = \frac{28 \pm 12\sqrt{10}}{8} = \frac{2(14 \pm 6\sqrt{10})}{8} = \frac{14 \pm 6\sqrt{10}}{4} = \frac{7 \pm 3\sqrt{10}}{2}
$$

✔ Solutions: $x = \frac{7 + 3\sqrt{10}}{2}, \frac{7 - 3\sqrt{10}}{2}$

---

6) $x^2 + 17x + 63 = -9$

Move $-9$ to the left:

$$
x^2 + 17x + 63 + 9 = 0 \Rightarrow x^2 + 17x + 72 = 0
$$

- $a = 1$, $b = 17$, $c = 72$

$$
x = \frac{-17 \pm \sqrt{(17)^2 - 4(1)(72)}}{2(1)} = \frac{-17 \pm \sqrt{289 - 288}}{2} = \frac{-17 \pm \sqrt{1}}{2}
$$

$$
\sqrt{1} = 1 \Rightarrow x = \frac{-17 \pm 1}{2}
$$

- $x = \frac{-17 + 1}{2} = \frac{-16}{2} = -8$
- $x = \frac{-17 - 1}{2} = \frac{-18}{2} = -9$

✔ Solutions: $x = -8, -9$

---

7) $2x^2 + 20x + 48 = 0$

Divide entire equation by 2:

$$
x^2 + 10x + 24 = 0
$$

- $a = 1$, $b = 10$, $c = 24$

$$
x = \frac{-10 \pm \sqrt{(10)^2 - 4(1)(24)}}{2} = \frac{-10 \pm \sqrt{100 - 96}}{2} = \frac{-10 \pm \sqrt{4}}{2}
$$

$$
\sqrt{4} = 2 \Rightarrow x = \frac{-10 \pm 2}{2}
$$

- $x = \frac{-10 + 2}{2} = \frac{-8}{2} = -4$
- $x = \frac{-10 - 2}{2} = \frac{-12}{2} = -6$

✔ Solutions: $x = -4, -6$

---

8) $-4x^2 + 8x + 21 = 0$

Already in standard form.
- $a = -4$, $b = 8$, $c = 21$

$$
x = \frac{-8 \pm \sqrt{(8)^2 - 4(-4)(21)}}{2(-4)} = \frac{-8 \pm \sqrt{64 + 336}}{-8} = \frac{-8 \pm \sqrt{400}}{-8}
$$

$$
\sqrt{400} = 20 \Rightarrow x = \frac{-8 \pm 20}{-8}
$$

Now compute both:

- $x = \frac{-8 + 20}{-8} = \frac{12}{-8} = -\frac{3}{2}$
- $x = \frac{-8 - 20}{-8} = \frac{-28}{-8} = \frac{7}{2}$

✔ Solutions: $x = -\frac{3}{2}, \frac{7}{2}$

---

✔ Final Answers:

1. $y = 6, -8$
2. $x = 1, -\frac{2}{3}$
3. $x = 7, \frac{3}{2}$
4. $x = 7, -4$
5. $x = \frac{7 + 3\sqrt{10}}{2}, \frac{7 - 3\sqrt{10}}{2}$
6. $x = -8, -9$
7. $x = -4, -6$
8. $x = -\frac{3}{2}, \frac{7}{2}$

Let me know if you'd like these written neatly in boxed format!