Quadratic Formula Worksheets (printable, online, answers, examples) - Free Printable

To solve each quadratic equation using the quadratic formula, we use the general form of a quadratic equation:

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

The quadratic formula is given by:

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

We will apply this formula to each equation step by step.

---

1. Solve \( 3g^2 - 17g - 6 = 0 \)

Here, \( a = 3 \), \( b = -17 \), and \( c = -6 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (-17)^2 - 4(3)(-6)
$$
$$
\Delta = 289 + 72 = 361
$$

#### Step 2: Apply the quadratic formula
$$
g = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-17) \pm \sqrt{361}}{2(3)}
$$
$$
g = \frac{17 \pm 19}{6}
$$

#### Step 3: Solve for the two roots
$$
g_1 = \frac{17 + 19}{6} = \frac{36}{6} = 6
$$
$$
g_2 = \frac{17 - 19}{6} = \frac{-2}{6} = -\frac{1}{3}
$$

#### Final Answer:
$$
g = 6 \quad \text{or} \quad g = -\frac{1}{3}
$$

---

2. Solve \( 4r^2 - 23r + 33 = 0 \)

Here, \( a = 4 \), \( b = -23 \), and \( c = 33 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (-23)^2 - 4(4)(33)
$$
$$
\Delta = 529 - 528 = 1
$$

#### Step 2: Apply the quadratic formula
$$
r = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-23) \pm \sqrt{1}}{2(4)}
$$
$$
r = \frac{23 \pm 1}{8}
$$

#### Step 3: Solve for the two roots
$$
r_1 = \frac{23 + 1}{8} = \frac{24}{8} = 3
$$
$$
r_2 = \frac{23 - 1}{8} = \frac{22}{8} = \frac{11}{4}
$$

#### Final Answer:
$$
r = 3 \quad \text{or} \quad r = \frac{11}{4}
$$

---

3. Solve \( 24x^2 + 22x - 7 = 0 \)

Here, \( a = 24 \), \( b = 22 \), and \( c = -7 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (22)^2 - 4(24)(-7)
$$
$$
\Delta = 484 + 672 = 1156
$$

#### Step 2: Apply the quadratic formula
$$
x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-22 \pm \sqrt{1156}}{2(24)}
$$
$$
x = \frac{-22 \pm 34}{48}
$$

#### Step 3: Solve for the two roots
$$
x_1 = \frac{-22 + 34}{48} = \frac{12}{48} = \frac{1}{4}
$$
$$
x_2 = \frac{-22 - 34}{48} = \frac{-56}{48} = -\frac{7}{6}
$$

#### Final Answer:
$$
x = \frac{1}{4} \quad \text{or} \quad x = -\frac{7}{6}
$$

---

4. Solve \( 2h^2 - 7h + 8 = 0 \)

Here, \( a = 2 \), \( b = -7 \), and \( c = 8 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (-7)^2 - 4(2)(8)
$$
$$
\Delta = 49 - 64 = -15
$$

#### Step 2: Analyze the discriminant
Since \( \Delta < 0 \), the quadratic equation has no real solutions. The solutions are complex.

#### Final Answer:
$$
\text{No real solutions}
$$

---

5. Solve \( 20b^2 + 11b - 4 = 0 \)

Here, \( a = 20 \), \( b = 11 \), and \( c = -4 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (11)^2 - 4(20)(-4)
$$
$$
\Delta = 121 + 320 = 441
$$

#### Step 2: Apply the quadratic formula
$$
b = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-11 \pm \sqrt{441}}{2(20)}
$$
$$
b = \frac{-11 \pm 21}{40}
$$

#### Step 3: Solve for the two roots
$$
b_1 = \frac{-11 + 21}{40} = \frac{10}{40} = \frac{1}{4}
$$
$$
b_2 = \frac{-11 - 21}{40} = \frac{-32}{40} = -\frac{4}{5}
$$

#### Final Answer:
$$
b = \frac{1}{4} \quad \text{or} \quad b = -\frac{4}{5}
$$

---

6. Solve \( 12g^2 + 41g + 35 = 0 \)

Here, \( a = 12 \), \( b = 41 \), and \( c = 35 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (41)^2 - 4(12)(35)
$$
$$
\Delta = 1681 - 1680 = 1
$$

#### Step 2: Apply the quadratic formula
$$
g = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-41 \pm \sqrt{1}}{2(12)}
$$
$$
g = \frac{-41 \pm 1}{24}
$$

#### Step 3: Solve for the two roots
$$
g_1 = \frac{-41 + 1}{24} = \frac{-40}{24} = -\frac{5}{3}
$$
$$
g_2 = \frac{-41 - 1}{24} = \frac{-42}{24} = -\frac{7}{4}
$$

#### Final Answer:
$$
g = -\frac{5}{3} \quad \text{or} \quad g = -\frac{7}{4}
$$

---

7. Solve \( 5n^2 + 19n + 18 = 0 \)

Here, \( a = 5 \), \( b = 19 \), and \( c = 18 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (19)^2 - 4(5)(18)
$$
$$
\Delta = 361 - 360 = 1
$$

#### Step 2: Apply the quadratic formula
$$
n = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-19 \pm \sqrt{1}}{2(5)}
$$
$$
n = \frac{-19 \pm 1}{10}
$$

#### Step 3: Solve for the two roots
$$
n_1 = \frac{-19 + 1}{10} = \frac{-18}{10} = -\frac{9}{5}
$$
$$
n_2 = \frac{-19 - 1}{10} = \frac{-20}{10} = -2
$$

#### Final Answer:
$$
n = -\frac{9}{5} \quad \text{or} \quad n = -2
$$

---

8. Solve \( 6n^2 - 17n = 14 \)

First, rewrite the equation in standard form:
$$
6n^2 - 17n - 14 = 0
$$

Here, \( a = 6 \), \( b = -17 \), and \( c = -14 \).

#### Step 1: Calculate the discriminant (\( \Delta \))
$$
\Delta = b^2 - 4ac = (-17)^2 - 4(6)(-14)
$$
$$
\Delta = 289 + 336 = 625
$$

#### Step 2: Apply the quadratic formula
$$
n = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-17) \pm \sqrt{625}}{2(6)}
$$
$$
n = \frac{17 \pm 25}{12}
$$

#### Step 3: Solve for the two roots
$$
n_1 = \frac{17 + 25}{12} = \frac{42}{12} = \frac{7}{2}
$$
$$
n_2 = \frac{17 - 25}{12} = \frac{-8}{12} = -\frac{2}{3}
$$

#### Final Answer:
$$
n = \frac{7}{2} \quad \text{or} \quad n = -\frac{2}{3}
$$

---

Final Answers:

1. \( g = 6 \) or \( g = -\frac{1}{3} \)
2. \( r = 3 \) or \( r = \frac{11}{4} \)
3. \( x = \frac{1}{4} \) or \( x = -\frac{7}{6} \)
4. No real solutions
5. \( b = \frac{1}{4} \) or \( b = -\frac{4}{5} \)
6. \( g = -\frac{5}{3} \) or \( g = -\frac{7}{4} \)
7. \( n = -\frac{9}{5} \) or \( n = -2 \)
8. \( n = \frac{7}{2} \) or \( n = -\frac{2}{3} \)

$$
\boxed{
\begin{aligned}
1. & \ g = 6 \ \text{or} \ g = -\frac{1}{3} \\
2. & \ r = 3 \ \text{or} \ r = \frac{11}{4} \\
3. & \ x = \frac{1}{4} \ \text{or} \ x = -\frac{7}{6} \\
4. & \ \text{No real solutions} \\
5. & \ b = \frac{1}{4} \ \text{or} \ b = -\frac{4}{5} \\
6. & \ g = -\frac{5}{3} \ \text{or} \ g = -\frac{7}{4} \\
7. & \ n = -\frac{9}{5} \ \text{or} \ n = -2 \\
8. & \ n = \frac{7}{2} \ \text{or} \ n = -\frac{2}{3}
\end{aligned}
}
$$