Solving Quadratic Equations Using The Quadratic Formula Worksheet - Free Printable

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

The quadratic formula is:

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

where the quadratic equation is in the form:

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

We'll go through each problem step by step.

---

1) $3n^2 - 5n - 8 = 0$

- $a = 3$, $b = -5$, $c = -8$

$$
n = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(-8)}}{2(3)} = \frac{5 \pm \sqrt{25 + 96}}{6} = \frac{5 \pm \sqrt{121}}{6}
$$

$$
\sqrt{121} = 11 \Rightarrow n = \frac{5 \pm 11}{6}
$$

- $n = \frac{5 + 11}{6} = \frac{16}{6} = \frac{8}{3}$
- $n = \frac{5 - 11}{6} = \frac{-6}{6} = -1$

✔ Solutions: $n = \frac{8}{3}, -1$

---

2) $x^2 + 10x + 21 = 0$

- $a = 1$, $b = 10$, $c = 21$

$$
x = \frac{-10 \pm \sqrt{10^2 - 4(1)(21)}}{2(1)} = \frac{-10 \pm \sqrt{100 - 84}}{2} = \frac{-10 \pm \sqrt{16}}{2}
$$

$$
\sqrt{16} = 4 \Rightarrow x = \frac{-10 \pm 4}{2}
$$

- $x = \frac{-10 + 4}{2} = \frac{-6}{2} = -3$
- $x = \frac{-10 - 4}{2} = \frac{-14}{2} = -7$

✔ Solutions: $x = -3, -7$

---

3) $10x^2 - 9x + 6 = 0$

- $a = 10$, $b = -9$, $c = 6$

$$
x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(10)(6)}}{2(10)} = \frac{9 \pm \sqrt{81 - 240}}{20} = \frac{9 \pm \sqrt{-159}}{20}
$$

Discriminant is negative → No real solutions

✔ Solutions: No real solutions (complex: $\frac{9 \pm i\sqrt{159}}{20}$)

---

4) $p^2 - 9 = 0$

Rewrite: $p^2 + 0p - 9 = 0$

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

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

✔ Solutions: $p = 3, -3$

---

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

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

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

$$
\sqrt{120} = \sqrt{4 \cdot 30} = 2\sqrt{30}
\Rightarrow x = \frac{12 \pm 2\sqrt{30}}{12} = \frac{6 \pm \sqrt{30}}{6}
$$

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

---

6) $6n^2 - 11 = 0$

Rewrite: $6n^2 + 0n - 11 = 0$

- $a = 6$, $b = 0$, $c = -11$

$$
n = \frac{-0 \pm \sqrt{0 - 4(6)(-11)}}{2(6)} = \frac{\pm \sqrt{264}}{12}
$$

$$
\sqrt{264} = \sqrt{4 \cdot 66} = 2\sqrt{66} \Rightarrow n = \frac{\pm 2\sqrt{66}}{12} = \frac{\pm \sqrt{66}}{6}
$$

✔ Solutions: $n = \pm \frac{\sqrt{66}}{6}$

---

7) $2n^2 + 5n - 9 = 0$

- $a = 2$, $b = 5$, $c = -9$

$$
n = \frac{-5 \pm \sqrt{5^2 - 4(2)(-9)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 72}}{4} = \frac{-5 \pm \sqrt{97}}{4}
$$

✔ Solutions: $n = \frac{-5 \pm \sqrt{97}}{4}$

---

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

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

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

$$
\sqrt{312} = \sqrt{4 \cdot 78} = 2\sqrt{78} \Rightarrow x = \frac{6 \pm 2\sqrt{78}}{6} = \frac{3 \pm \sqrt{78}}{3}
$$

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

---

9) $6k^2 + 12k - 15 = -10$

Bring all terms to one side:

$$
6k^2 + 12k - 15 + 10 = 0 \Rightarrow 6k^2 + 12k - 5 = 0
$$

- $a = 6$, $b = 12$, $c = -5$

$$
k = \frac{-12 \pm \sqrt{12^2 - 4(6)(-5)}}{2(6)} = \frac{-12 \pm \sqrt{144 + 120}}{12} = \frac{-12 \pm \sqrt{264}}{12}
$$

$$
\sqrt{264} = 2\sqrt{66} \Rightarrow k = \frac{-12 \pm 2\sqrt{66}}{12} = \frac{-6 \pm \sqrt{66}}{6}
$$

✔ Solutions: $k = \frac{-6 \pm \sqrt{66}}{6}$

---

10) $8x^2 - 14 = -11$

Bring to standard form:

$$
8x^2 - 14 + 11 = 0 \Rightarrow 8x^2 - 3 = 0
$$

- $a = 8$, $b = 0$, $c = -3$

$$
x = \frac{\pm \sqrt{0 - 4(8)(-3)}}{2(8)} = \frac{\pm \sqrt{96}}{16} = \frac{\pm \sqrt{16 \cdot 6}}{16} = \frac{\pm 4\sqrt{6}}{16} = \frac{\pm \sqrt{6}}{4}
$$

✔ Solutions: $x = \pm \frac{\sqrt{6}}{4}$

---

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

Bring to standard form:

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

Divide all terms by 2: $3k^2 + k + 6 = 0$

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

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

No real solutions

✔ Solutions: No real solutions

---

12) $12p^2 + 9p - 30 = -10$

Bring to standard form:

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

- $a = 12$, $b = 9$, $c = -20$

$$
p = \frac{-9 \pm \sqrt{9^2 - 4(12)(-20)}}{2(12)} = \frac{-9 \pm \sqrt{81 + 960}}{24} = \frac{-9 \pm \sqrt{1041}}{24}
$$

$\sqrt{1041}$ cannot be simplified

✔ Solutions: $p = \frac{-9 \pm \sqrt{1041}}{24}$

---

13) $3x^2 = -7x + 136$

Bring all terms to left:

$$
3x^2 + 7x - 136 = 0
$$

- $a = 3$, $b = 7$, $c = -136$

$$
x = \frac{-7 \pm \sqrt{7^2 - 4(3)(-136)}}{2(3)} = \frac{-7 \pm \sqrt{49 + 1632}}{6} = \frac{-7 \pm \sqrt{1681}}{6}
$$

$$
\sqrt{1681} = 41 \Rightarrow x = \frac{-7 \pm 41}{6}
$$

- $x = \frac{-7 + 41}{6} = \frac{34}{6} = \frac{17}{3}$
- $x = \frac{-7 - 41}{6} = \frac{-48}{6} = -8$

✔ Solutions: $x = \frac{17}{3}, -8$

---

14) $3n^2 = -n + 14$

Bring all terms to left:

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

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

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

$$
\sqrt{169} = 13 \Rightarrow n = \frac{-1 \pm 13}{6}
$$

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

✔ Solutions: $n = 2, -\frac{7}{3}$

---

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

Bring all terms to left:

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

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

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

No real solutions

✔ Solutions: No real solutions

---

16) $9p^2 - 7 = 9p$

Bring to standard form:

$$
9p^2 - 9p - 7 = 0
$$

- $a = 9$, $b = -9$, $c = -7$

$$
p = \frac{-(-9) \pm \sqrt{(-9)^2 - 4(9)(-7)}}{2(9)} = \frac{9 \pm \sqrt{81 + 252}}{18} = \frac{9 \pm \sqrt{333}}{18}
$$

$$
\sqrt{333} = \sqrt{9 \cdot 37} = 3\sqrt{37} \Rightarrow p = \frac{9 \pm 3\sqrt{37}}{18} = \frac{3 \pm \sqrt{37}}{6}
$$

✔ Solutions: $p = \frac{3 \pm \sqrt{37}}{6}$

---

17) $11k^2 + 4k - 52 = 10k^2 - 7$

Bring all terms to left:

$$
11k^2 - 10k^2 + 4k - 52 + 7 = 0 \Rightarrow k^2 + 4k - 45 = 0
$$

- $a = 1$, $b = 4$, $c = -45$

$$
k = \frac{-4 \pm \sqrt{4^2 - 4(1)(-45)}}{2(1)} = \frac{-4 \pm \sqrt{16 + 180}}{2} = \frac{-4 \pm \sqrt{196}}{2}
$$

$$
\sqrt{196} = 14 \Rightarrow k = \frac{-4 \pm 14}{2}
$$

- $k = \frac{-4 + 14}{2} = \frac{10}{2} = 5$
- $k = \frac{-4 - 14}{2} = \frac{-18}{2} = -9$

✔ Solutions: $k = 5, -9$

---

18) $-4a^2 + 18a - 15 = -7a^2 + 9a$

Bring all terms to left:

$$
-4a^2 + 18a - 15 + 7a^2 - 9a = 0 \Rightarrow 3a^2 + 9a - 15 = 0
$$

Divide by 3: $a^2 + 3a - 5 = 0$

- $a = 1$, $b = 3$, $c = -5$

$$
a = \frac{-3 \pm \sqrt{3^2 - 4(1)(-5)}}{2(1)} = \frac{-3 \pm \sqrt{9 + 20}}{2} = \frac{-3 \pm \sqrt{29}}{2}
$$

✔ Solutions: $a = \frac{-3 \pm \sqrt{29}}{2}$

---

19) $-4n(n - 2) = 6(n + 3) - 11n^2$

Expand both sides:

Left: $-4n(n - 2) = -4n^2 + 8n$

Right: $6n + 18 - 11n^2 = -11n^2 + 6n + 18$

Now bring all terms to left:

$$
-4n^2 + 8n + 11n^2 - 6n - 18 = 0 \Rightarrow 7n^2 + 2n - 18 = 0
$$

- $a = 7$, $b = 2$, $c = -18$

$$
n = \frac{-2 \pm \sqrt{2^2 - 4(7)(-18)}}{2(7)} = \frac{-2 \pm \sqrt{4 + 504}}{14} = \frac{-2 \pm \sqrt{508}}{14}
$$

$$
\sqrt{508} = \sqrt{4 \cdot 127} = 2\sqrt{127} \Rightarrow n = \frac{-2 \pm 2\sqrt{127}}{14} = \frac{-1 \pm \sqrt{127}}{7}
$$

✔ Solutions: $n = \frac{-1 \pm \sqrt{127}}{7}$

---

20) $x(x - 3) = -7 - 10x$

Expand left: $x^2 - 3x$

Right: $-7 - 10x$

Bring all to left:

$$
x^2 - 3x + 10x + 7 = 0 \Rightarrow x^2 + 7x + 7 = 0
$$

- $a = 1$, $b = 7$, $c = 7$

$$
x = \frac{-7 \pm \sqrt{7^2 - 4(1)(7)}}{2(1)} = \frac{-7 \pm \sqrt{49 - 28}}{2} = \frac{-7 \pm \sqrt{21}}{2}
$$

✔ Solutions: $x = \frac{-7 \pm \sqrt{21}}{2}$

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## ✔ Final Answers Summary:

| Problem | Solutions |
|--------|-----------|
| 1 | $n = \frac{8}{3}, -1$ |
| 2 | $x = -3, -7$ |
| 3 | No real solutions |
| 4 | $p = 3, -3$ |
| 5 | $x = \frac{6 \pm \sqrt{30}}{6}$ |
| 6 | $n = \pm \frac{\sqrt{66}}{6}$ |
| 7 | $n = \frac{-5 \pm \sqrt{97}}{4}$ |
| 8 | $x = \frac{3 \pm \sqrt{78}}{3}$ |
| 9 | $k = \frac{-6 \pm \sqrt{66}}{6}$ |
| 10 | $x = \pm \frac{\sqrt{6}}{4}$ |
| 11 | No real solutions |
| 12 | $p = \frac{-9 \pm \sqrt{1041}}{24}$ |
| 13 | $x = \frac{17}{3}, -8$ |
| 14 | $n = 2, -\frac{7}{3}$ |
| 15 | No real solutions |
| 16 | $p = \frac{3 \pm \sqrt{37}}{6}$ |
| 17 | $k = 5, -9$ |
| 18 | $a = \frac{-3 \pm \sqrt{29}}{2}$ |
| 19 | $n = \frac{-1 \pm \sqrt{127}}{7}$ |
| 20 | $x = \frac{-7 \pm \sqrt{21}}{2}$ |

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