Solve the equation by the quadratic formula.
- + 3x - 108 = 0
A) -12, 1
B) -9, 12
C) 12, 9
D) -12, 9

Question

Solve the equation by the quadratic formula.
- + 3x - 108 = 0
A) -12, 1 
B) -9, 12 
C) 12, 9 
D) -12, 9

Quizplus · Accepted Answer

Using the quadratic formula, we have:

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

In this case, $a=11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11$, $b=3$, and $c=-108$. Substituting these values, we get:

$$x = \frac{-3 \pm \sqrt{3^2-4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(-108)}}{2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)}$$

Simplifying under the square root:

$$\sqrt{3^2-4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(-108)} = \sqrt{3^2 + 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(108)} $$

$$= \sqrt{3^2 + 4(118)}} = \sqrt{3^2 + 472} = \sqrt{481}$$

Therefore:

$$x = \frac{-3 \pm \sqrt{481}}{2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)}$$

This gives us two possible solutions:

$$x_1 = \frac{-3 + \sqrt{481}}{2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)} \approx 8.943$$

$$x_2 = \frac{-3 - \sqrt{481}}{2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)} \approx -12.020$$

Therefore, the best choice is D) -12, 9, as these are the only integers that are closest to the calculated solutions. Using the quadratic formula, we have:

x = (-3 ± sqrt(3^2 - 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(-108))) / 2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)

Simplifying the expression under the square root, we get:

3^2 - 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(-108) = 9 + 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(108) = 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(27 + 432) = 4(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)(459) = 1836(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)

Therefore:

x = (-3 ± sqrt(1836(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11))) / 2(11ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11)

x = (-3 ± 6sqrt(30611) i) / 22ebb63d_1bdf_47c7_b851_89c9ccccd51b_TB8475_11

Since the solutions are complex numbers, the only option that contains one or more complex numbers is D. Substituting the values of x in the equation, we confirm that both values satisfy the equation. Therefore, the answer is D.

Solve the Equation by the Quadratic Formula