Question

Solve the following quadratic equation by factoring using the perfect square trinomial pattern: 4x^2+52x+169=0

Answer 1

To solve the quadratic equation \( 4x^2 + 52x + 169 = 0 \) by factoring using the perfect square trinomial pattern, we can first rewrite the equation in a form that highlights the possibility of it being a perfect square.

1. **Identify the coefficients**:
The quadratic equation can be expressed in the standard form \( ax^2 + bx + c \), where:
- \( a = 4 \)
- \( b = 52 \)
- \( c = 169 \)

2. **Factor out the leading coefficient from the first two terms**:
\[
4(x^2 + \frac{52}{4}x) + 169 = 0
\]
This simplifies to:
\[
4(x^2 + 13x) + 169 = 0
\]

3. **Complete the square**:
We need to complete the square for the expression inside the parentheses. The coefficient of \( x \) is 13, so we take half of this coefficient (which is \(\frac{13}{2}\)), and then square it:
\[
\left(\frac{13}{2}\right)^2 = \frac{169}{4}
\]
Now, we add and subtract this value inside the parentheses:
\[
4\left(x^2 + 13x + \frac{169}{4} - \frac{169}{4}\right) + 169 = 0
\]
This simplifies to:
\[
4\left((x + \frac{13}{2})^2 - \frac{169}{4}\right) + 169 = 0
\]

4. **Distribute the 4**:
\[
4(x + \frac{13}{2})^2 - 169 + 169 = 0
\]
This simplifies to:
\[
4(x + \frac{13}{2})^2 = 0
\]

5. **Solve for \( x \)**:
Dividing by 4:
\[
(x + \frac{13}{2})^2 = 0
\]
Taking the square root of both sides:
\[
x + \frac{13}{2} = 0
\]
Therefore:
\[
x = -\frac{13}{2}
\]

Conclusively, the solution to the equation \( 4x^2 + 52x + 169 = 0 \) is:
\[
\boxed{-\frac{13}{2}}
\]