Question

Solve the following quadratic equation by factoring using the perfect square trinomial pattern: 4x2+52x+169=0
.(1 point)
Responses

x=6.5
x equals 6.5

x=−8.45
x equals negative 8.45

x=−6.5
x equals negative 6.5

x=8.45

Answer 1

To solve the quadratic equation \(4x^2 + 52x + 169 = 0\) by factoring, we need to check if it can be expressed as a perfect square trinomial. A perfect square trinomial has the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\).

Let's start by rewriting the equation:

1. **Factor out the coefficient of \(x^2\)** (which is 4 in this case):

\[
4(x^2 + 13x) + 169 = 0
\]

2. **Now, rearrange the equation**:

\[
4(x^2 + 13x + \text{something}) + 169 - \text{something} = 0
\]

We can complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of \(x\) (which is 13), square it, and add it inside:

Half of 13 is \( \frac{13}{2} \) and squaring it gives:

\[
\left(\frac{13}{2}\right)^2 = \frac{169}{4}
\]

Now we can express this trinomial as a perfect square:

\[
x^2 + 13x + \frac{169}{4} = \left(x + \frac{13}{2}\right)^2
\]

Therefore, we rewrite our original equation after completing the square:

\[
4\left(x + \frac{13}{2}\right)^2 - 4 \cdot \frac{169}{4} + 169 = 0
\]

Simplifying gives:

\[
4\left(x + \frac{13}{2}\right)^2 = 0
\]

3. **Set the equation to zero**:

\[
\left(x + \frac{13}{2}\right)^2 = 0
\]

Taking the square root of both sides, we find:

\[
x + \frac{13}{2} = 0
\]

So,

\[
x = -\frac{13}{2} = -6.5
\]

The solution to the quadratic equation \(4x^2 + 52x + 169 = 0\) is \(x = -6.5\).

**Answer**: \(x = -6.5\)