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=−6.5 x equals negative 6.5 x=−8.45 x equals negative 8.45 x=8.45 x equals 8.45

Answer 1

To solve the quadratic equation \( 4x^2 + 52x + 169 = 0 \) by factoring using the perfect square trinomial pattern, we start by identifying if the expression can be represented as a square of a binomial.

The given equation is in the standard form \( ax^2 + bx + c \). We can factor it as follows:

1. First, identify \( a \), \( b \), and \( c \):
- \( a = 4 \)
- \( b = 52 \)
- \( c = 169 \)

2. Next, check if the equation can be written as a perfect square trinomial. A perfect square trinomial has the form \( (px + q)^2 = p^2x^2 + 2pqx + q^2 \).

3. The expression \( 4x^2 + 52x + 169 \) can be rewritten as:
\[
(2x)^2 + 2(2x)(26) + 26^2
\]
Which gives us:
\[
4x^2 + 104x + 676 = 0
\]

4. Since \( 4x^2 + 52x + 169 \) has a middle coefficient of \( 52 \), we can also directly express it:
Let's check if the middle term matches \( 2(2x)(13) \):
\[
(2x + 13)^2 = 4x^2 + 52x + 169
\]

Thus, we can write:
\[
(2x + 13)^2 = 0
\]

5. From this equation:
\[
2x + 13 = 0
\]
Solving for \( x \):
\[
2x = -13 \quad \Rightarrow \quad x = -\frac{13}{2} = -6.5
\]

So, the solutions of the equation \( 4x^2 + 52x + 169 = 0 \) are:
\[
x = -6.5
\]

Thus, the correct response is:
**x = -6.5**