Solving Quadratic Equations Using The Quadratic Formula Worksheet - Free Printable

It looks like you've uploaded a worksheet titled "Solving Quadratic Equations Using the Quadratic Formula" with 20 quadratic equations to solve. Since I can't view images directly, I’ll help you solve each equation step-by-step using the quadratic formula, which is:

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

This formula applies to any quadratic equation in the form:

$$
ax^2 + bx + c = 0
$$

Let’s go through each problem one by one and solve it.

---

1) $3x^2 - 9x - 8 = 0$

- $a = 3$, $b = -9$, $c = -8$

$$
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(3)(-8)}}{2(3)} = \frac{9 \pm \sqrt{81 + 96}}{6} = \frac{9 \pm \sqrt{177}}{6}
$$

✔ Answer: $x = \frac{9 \pm \sqrt{177}}{6}$

---

2) $x^2 + 10x + 25 = 0$

- $a = 1$, $b = 10$, $c = 25$

$$
x = \frac{-10 \pm \sqrt{10^2 - 4(1)(25)}}{2(1)} = \frac{-10 \pm \sqrt{100 - 100}}{2} = \frac{-10 \pm \sqrt{0}}{2} = \frac{-10}{2} = -5
$$

✔ Answer: $x = -5$ (double root)

---

3) $20x^2 - 36x + 6 = 0$

We can simplify first: divide all terms by 2:

$$
10x^2 - 18x + 3 = 0
$$

Now use formula: $a = 10$, $b = -18$, $c = 3$

$$
x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(10)(3)}}{2(10)} = \frac{18 \pm \sqrt{324 - 120}}{20} = \frac{18 \pm \sqrt{204}}{20}
$$

Simplify $\sqrt{204} = \sqrt{4 \cdot 51} = 2\sqrt{51}$

$$
x = \frac{18 \pm 2\sqrt{51}}{20} = \frac{9 \pm \sqrt{51}}{10}
$$

✔ Answer: $x = \frac{9 \pm \sqrt{51}}{10}$

---

4) $x^2 - 9 = 0$

This is easy: $x^2 = 9$, so $x = \pm 3$

But let's use the formula:

- $a = 1$, $b = 0$, $c = -9$

$$
x = \frac{-0 \pm \sqrt{0^2 - 4(1)(-9)}}{2(1)} = \frac{\pm \sqrt{36}}{2} = \frac{\pm 6}{2} = \pm 3
$$

✔ Answer: $x = \pm 3$

---

5) $6x^2 + 12x + 1 = 0$

- $a = 6$, $b = 12$, $c = 1$

$$
x = \frac{-12 \pm \sqrt{12^2 - 4(6)(1)}}{2(6)} = \frac{-12 \pm \sqrt{144 - 24}}{12} = \frac{-12 \pm \sqrt{120}}{12}
$$

$\sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30}$

$$
x = \frac{-12 \pm 2\sqrt{30}}{12} = \frac{-6 \pm \sqrt{30}}{6}
$$

✔ Answer: $x = \frac{-6 \pm \sqrt{30}}{6}$

---

6) $x^2 + 12 = 0$

- $a = 1$, $b = 0$, $c = 12$

$$
x = \frac{-0 \pm \sqrt{0 - 4(1)(12)}}{2} = \frac{\pm \sqrt{-48}}{2} = \frac{\pm 4i\sqrt{3}}{2} = \pm 2i\sqrt{3}
$$

✔ Answer: $x = \pm 2i\sqrt{3}$ (complex roots)

---

7) $2x^2 + 6x - 9 = 0$

- $a = 2$, $b = 6$, $c = -9$

$$
x = \frac{-6 \pm \sqrt{6^2 - 4(2)(-9)}}{2(2)} = \frac{-6 \pm \sqrt{36 + 72}}{4} = \frac{-6 \pm \sqrt{108}}{4}
$$

$\sqrt{108} = \sqrt{36 \cdot 3} = 6\sqrt{3}$

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

✔ Answer: $x = \frac{-3 \pm 3\sqrt{3}}{2}$

---

8) $3x^2 - 6x - 23 = 0$

- $a = 3$, $b = -6$, $c = -23$

$$
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(3)(-23)}}{2(3)} = \frac{6 \pm \sqrt{36 + 276}}{6} = \frac{6 \pm \sqrt{312}}{6}
$$

$\sqrt{312} = \sqrt{4 \cdot 78} = 2\sqrt{78}$

$$
x = \frac{6 \pm 2\sqrt{78}}{6} = \frac{3 \pm \sqrt{78}}{3}
$$

✔ Answer: $x = \frac{3 \pm \sqrt{78}}{3}$

---

9) $6x^2 + 12x - 14 = -20$

First, bring all terms to one side:

$$
6x^2 + 12x - 14 + 20 = 0 \Rightarrow 6x^2 + 12x + 6 = 0
$$

Divide by 6: $x^2 + 2x + 1 = 0$

Now: $a = 1$, $b = 2$, $c = 1$

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

✔ Answer: $x = -1$ (double root)

---

10) $x^2 - 14 = -11$

Bring to standard form:

$$
x^2 - 14 + 11 = 0 \Rightarrow x^2 - 3 = 0
$$

So $x^2 = 3$, $x = \pm \sqrt{3}$

Using formula: $a = 1$, $b = 0$, $c = -3$

$$
x = \frac{0 \pm \sqrt{0 + 12}}{2} = \frac{\pm \sqrt{12}}{2} = \frac{\pm 2\sqrt{3}}{2} = \pm \sqrt{3}
$$

✔ Answer: $x = \pm \sqrt{3}$

---

11) $6x^2 + 2x + 9 = -3$

Move all terms:

$$
6x^2 + 2x + 9 + 3 = 0 \Rightarrow 6x^2 + 2x + 12 = 0
$$

Divide by 2: $3x^2 + x + 6 = 0$

$a = 3$, $b = 1$, $c = 6$

$$
x = \frac{-1 \pm \sqrt{1^2 - 4(3)(6)}}{2(3)} = \frac{-1 \pm \sqrt{1 - 72}}{6} = \frac{-1 \pm \sqrt{-71}}{6}
$$

✔ Answer: $x = \frac{-1 \pm i\sqrt{71}}{6}$ (complex)

---

12) $12x^2 + 9x - 30 = -20$

Bring to zero:

$$
12x^2 + 9x - 30 + 20 = 0 \Rightarrow 12x^2 + 9x - 10 = 0
$$

$a = 12$, $b = 9$, $c = -10$

$$
x = \frac{-9 \pm \sqrt{81 - 4(12)(-10)}}{2(12)} = \frac{-9 \pm \sqrt{81 + 480}}{24} = \frac{-9 \pm \sqrt{561}}{24}
$$

$\sqrt{561}$ doesn’t simplify (no perfect square factors), so leave as is.

✔ Answer: $x = \frac{-9 \pm \sqrt{561}}{24}$

---

13) $3x^2 - 7x = 126$

Bring to zero:

$$
3x^2 - 7x - 126 = 0
$$

$a = 3$, $b = -7$, $c = -126$

$$
x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(3)(-126)}}{2(3)} = \frac{7 \pm \sqrt{49 + 1512}}{6} = \frac{7 \pm \sqrt{1561}}{6}
$$

Check if 1561 simplifies: 1561 ÷ 7 ≈ 223 → not divisible by small squares. So leave as is.

✔ Answer: $x = \frac{7 \pm \sqrt{1561}}{6}$

---

14) $3x^2 + x = 14$

Bring to zero:

$$
3x^2 + x - 14 = 0
$$

$a = 3$, $b = 1$, $c = -14$

$$
x = \frac{-1 \pm \sqrt{1^2 - 4(3)(-14)}}{2(3)} = \frac{-1 \pm \sqrt{1 + 168}}{6} = \frac{-1 \pm \sqrt{169}}{6} = \frac{-1 \pm 13}{6}
$$

So:

- $x = \frac{-1 + 13}{6} = \frac{12}{6} = 2$
- $x = \frac{-1 - 13}{6} = \frac{-14}{6} = -\frac{7}{3}$

✔ Answer: $x = 2$, $x = -\frac{7}{3}$

---

15) $6x^2 + 3 = -2x$

Bring to zero:

$$
6x^2 + 2x + 3 = 0
$$

$a = 6$, $b = 2$, $c = 3$

$$
x = \frac{-2 \pm \sqrt{4 - 4(6)(3)}}{12} = \frac{-2 \pm \sqrt{4 - 72}}{12} = \frac{-2 \pm \sqrt{-68}}{12}
$$

$\sqrt{-68} = 2i\sqrt{17}$

$$
x = \frac{-2 \pm 2i\sqrt{17}}{12} = \frac{-1 \pm i\sqrt{17}}{6}
$$

✔ Answer: $x = \frac{-1 \pm i\sqrt{17}}{6}$

---

16) $9x^2 - 7 = 0$

Rewrite: $9x^2 = 7$, $x^2 = \frac{7}{9}$, $x = \pm \frac{\sqrt{7}}{3}$

Using formula: $a = 9$, $b = 0$, $c = -7$

$$
x = \frac{0 \pm \sqrt{0 + 252}}{18} = \frac{\pm \sqrt{252}}{18}
$$

$\sqrt{252} = \sqrt{36 \cdot 7} = 6\sqrt{7}$

$$
x = \frac{\pm 6\sqrt{7}}{18} = \pm \frac{\sqrt{7}}{3}
$$

✔ Answer: $x = \pm \frac{\sqrt{7}}{3}$

---

17) $12x^2 + 4x - 52 = 10x^2 - 7$

Bring all terms to one side:

$$
12x^2 - 10x^2 + 4x + 7 - 52 = 0 \Rightarrow 2x^2 + 4x - 45 = 0
$$

$a = 2$, $b = 4$, $c = -45$

$$
x = \frac{-4 \pm \sqrt{16 - 4(2)(-45)}}{4} = \frac{-4 \pm \sqrt{16 + 360}}{4} = \frac{-4 \pm \sqrt{376}}{4}
$$

$\sqrt{376} = \sqrt{4 \cdot 94} = 2\sqrt{94}$

$$
x = \frac{-4 \pm 2\sqrt{94}}{4} = \frac{-2 \pm \sqrt{94}}{2}
$$

✔ Answer: $x = \frac{-2 \pm \sqrt{94}}{2}$

---

18) $-4x^2 + 18x - 15 = -3x^2 + 6x$

Bring all to left:

$$
-4x^2 + 18x - 15 + 3x^2 - 6x = 0 \Rightarrow -x^2 + 12x - 15 = 0
$$

Multiply both sides by -1: $x^2 - 12x + 15 = 0$

$a = 1$, $b = -12$, $c = 15$

$$
x = \frac{12 \pm \sqrt{144 - 60}}{2} = \frac{12 \pm \sqrt{84}}{2} = \frac{12 \pm 2\sqrt{21}}{2} = 6 \pm \sqrt{21}
$$

✔ Answer: $x = 6 \pm \sqrt{21}$

---

19) $-46x - 2(-46)x + 8 = -12x^2$

Simplify left side:

$-46x + 92x + 8 = 46x + 8$

So: $46x + 8 = -12x^2$

Bring all to one side:

$$
12x^2 + 46x + 8 = 0
$$

Divide by 2: $6x^2 + 23x + 4 = 0$

$a = 6$, $b = 23$, $c = 4$

$$
x = \frac{-23 \pm \sqrt{529 - 96}}{12} = \frac{-23 \pm \sqrt{433}}{12}
$$

No simplification for $\sqrt{433}$

✔ Answer: $x = \frac{-23 \pm \sqrt{433}}{12}$

---

20) $x(x - 3) = -x + 20$

Left side: $x^2 - 3x$

Right side: $-x + 20$

Bring all to one side:

$$
x^2 - 3x + x - 20 = 0 \Rightarrow x^2 - 2x - 20 = 0
$$

$a = 1$, $b = -2$, $c = -20$

$$
x = \frac{2 \pm \sqrt{4 + 80}}{2} = \frac{2 \pm \sqrt{84}}{2} = \frac{2 \pm 2\sqrt{21}}{2} = 1 \pm \sqrt{21}
$$

✔ Answer: $x = 1 \pm \sqrt{21}$

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✔ Final Answers Summary:

| Problem | Answer |
|--------|--------|
| 1 | $x = \frac{9 \pm \sqrt{177}}{6}$ |
| 2 | $x = -5$ |
| 3 | $x = \frac{9 \pm \sqrt{51}}{10}$ |
| 4 | $x = \pm 3$ |
| 5 | $x = \frac{-6 \pm \sqrt{30}}{6}$ |
| 6 | $x = \pm 2i\sqrt{3}$ |
| 7 | $x = \frac{-3 \pm 3\sqrt{3}}{2}$ |
| 8 | $x = \frac{3 \pm \sqrt{78}}{3}$ |
| 9 | $x = -1$ |
| 10 | $x = \pm \sqrt{3}$ |
| 11 | $x = \frac{-1 \pm i\sqrt{71}}{6}$ |
| 12 | $x = \frac{-9 \pm \sqrt{561}}{24}$ |
| 13 | $x = \frac{7 \pm \sqrt{1561}}{6}$ |
| 14 | $x = 2$, $x = -\frac{7}{3}$ |
| 15 | $x = \frac{-1 \pm i\sqrt{17}}{6}$ |
| 16 | $x = \pm \frac{\sqrt{7}}{3}$ |
| 17 | $x = \frac{-2 \pm \sqrt{94}}{2}$ |
| 18 | $x = 6 \pm \sqrt{21}$ |
| 19 | $x = \frac{-23 \pm \sqrt{433}}{12}$ |
| 20 | $x = 1 \pm \sqrt{21}$ |

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