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 first need to recognize the structure of the equation.

1. **Rewrite the quadratic**: We can express the quadratic in the standard form \(ax^2 + bx + c\), where \(a = 4\), \(b = 52\), and \(c = 169\).

2. **Identify the perfect square**: We can factor out the coefficient of \(x^2\):
\[
4(x^2 + 13x) + 169 = 0
\]

3. **Complete the square**: To complete the square for the expression \(x^2 + 13x\):
- Take half of the coefficient of \(x\) (which is 13), square it: \((\frac{13}{2})^2 = \frac{169}{4}\).
- 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:
\[
4\left((x + \frac{13}{2})^2 - \frac{169}{4}\right) + 169 = 0
\]
- Distributing the \(4\) gives:
\[
4\left(x + \frac{13}{2}\right)^2 - 169 + 169 = 0
\]
- Therefore, we have:
\[
4\left(x + \frac{13}{2}\right)^2 = 0
\]

4. **Solve for \(x\)**:
- Set the squared term to zero:
\[
\left(x + \frac{13}{2}\right)^2 = 0
\]
- Taking the square root of both sides gives:
\[
x + \frac{13}{2} = 0
\]
- Solving for \(x\), we find:
\[
x = -\frac{13}{2}
\]

Thus, the solution to the equation \(4x^2 + 52x + 169 = 0\) is

\[
\boxed{-\frac{13}{2}}.
\]