Which of the following are solutions to the equation below?
Check all that apply.

[tex]\[ 4x^2 + 32x + 60 = 0 \][/tex]

A. -5
B. -3
C. 3
D. 5
E. -32



Answer :

To solve the quadratic equation [tex]\(4x^2 + 32x + 60 = 0\)[/tex] and determine which of the given choices are valid solutions, we can follow these steps:

### Step 1: Solve the quadratic equation using the quadratic formula

The quadratic formula is given by:

[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]

For the equation [tex]\(4x^2 + 32x + 60 = 0\)[/tex], the coefficients are:

- [tex]\(a = 4\)[/tex]
- [tex]\(b = 32\)[/tex]
- [tex]\(c = 60\)[/tex]

### Step 2: Calculate the discriminant

The discriminant ([tex]\(\Delta\)[/tex]) is calculated as follows:

[tex]\[ \Delta = b^2 - 4ac \][/tex]

Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:

[tex]\[ \Delta = (32)^2 - 4(4)(60) \][/tex]
[tex]\[ \Delta = 1024 - 960 \][/tex]
[tex]\[ \Delta = 64 \][/tex]

### Step 3: Calculate the solutions using the quadratic formula

Since the discriminant is positive ([tex]\(\Delta = 64\)[/tex]), there are two real roots. Substitute [tex]\(\Delta\)[/tex] into the quadratic formula:

[tex]\[ x = \frac{-32 \pm \sqrt{64}}{2 \cdot 4} \][/tex]
[tex]\[ x = \frac{-32 \pm 8}{8} \][/tex]

### Step 4: Simplify the two potential solutions

1. Using the positive square root:

[tex]\[ x = \frac{-32 + 8}{8} \][/tex]
[tex]\[ x = \frac{-24}{8} \][/tex]
[tex]\[ x = -3 \][/tex]

2. Using the negative square root:

[tex]\[ x = \frac{-32 - 8}{8} \][/tex]
[tex]\[ x = \frac{-40}{8} \][/tex]
[tex]\[ x = -5 \][/tex]

### Step 5: Evaluate the given choices

The solutions to the equation [tex]\(4x^2 + 32x + 60 = 0\)[/tex] are [tex]\(x = -3\)[/tex] and [tex]\(x = -5\)[/tex]. Thus, we can evaluate which of the given choices are valid solutions:

- A. -5 \<-- This is a solution.
- B. -3 \<-- This is a solution.
- C. 3 \<-- This is not a solution.
- D. 5 \<-- This is not a solution.
- E. -32 \<-- This is not a solution.

### Final Answer:

The solutions to the equation [tex]\(4x^2 + 32x + 60 = 0\)[/tex] from the given choices are:

- A. -5
- B. -3