To solve the quadratic equation [tex]\(9x^2 + 24x + 16 = 0\)[/tex] by factoring, let's follow these steps:
1. Write the equation in standard form:
The equation is already in standard form, which is [tex]\(ax^2 + bx + c = 0\)[/tex].
2. Identify the coefficients:
Here, [tex]\(a = 9\)[/tex], [tex]\(b = 24\)[/tex], and [tex]\(c = 16\)[/tex].
3. Factor the quadratic expression:
We need to express [tex]\(9x^2 + 24x + 16\)[/tex] as a product of two binomials of the form [tex]\((mx + n)(px + q)\)[/tex].
Our goal is to find two numbers that multiply to [tex]\(a \cdot c = 9 \cdot 16 = 144\)[/tex] and add up to [tex]\(b = 24\)[/tex].
Notice that the quadratic can be factored as:
[tex]\[
9x^2 + 24x + 16 = (3x + 4)(3x + 4)
\][/tex]
Explanation:
- [tex]\(3x \cdot 3x = 9x^2\)[/tex]
- [tex]\(3x \cdot 4 = 12x\)[/tex]
- [tex]\(4 \cdot 3x = 12x\)[/tex]
- [tex]\(4 \cdot 4 = 16\)[/tex]
When you combine the middle terms: [tex]\(12x + 12x = 24x\)[/tex].
So, [tex]\(9x^2 + 24x + 16 = (3x + 4)^2\)[/tex].
4. Set each factor to zero and solve for [tex]\(x\)[/tex]:
Since the quadratic can be written as [tex]\((3x + 4)^2 = 0\)[/tex], we set the inside of the square to zero:
[tex]\[
3x + 4 = 0
\][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
3x + 4 = 0 \implies 3x = -4 \implies x = -\frac{4}{3}
\][/tex]
5. Write the solution:
The solution to the quadratic equation [tex]\(9x^2 + 24x + 16 = 0\)[/tex] is:
[tex]\[
x = -\frac{4}{3}
\][/tex]
Therefore, the correct choice from the given options is [tex]\(x = -4 / 3\)[/tex]. The answer is:
```
[tex]\[
[-4/3]
\][/tex]
```
