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
### 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
