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

Question

.(1 point)
Responses

x=8.45
x equals 8.45

x=6.5
x equals 6.5

x=−8.45
x equals negative 8.45

x=−6.5

Answer 1

To solve the quadratic equation \( 4x^2 + 52x + 169 = 0 \) by factoring, we first look for a way to express it as a perfect square trinomial.

1. Observe the coefficients:

   • The leading coefficient is \( 4 \), which is \( (2)^2 \).
   • The constant term is \( 169 \), which is \( (13)^2 \).
   • The middle term \( 52x \) can be represented as \( 2 \cdot 2 \cdot 13 \cdot x \).

2. Rewrite the equation: Since \( 52 = 2 \times 2 \times 13 \), we can write: \[ 4x^2 + 52x + 169 = (2x + 13)^2 \]

3. Set the equation to zero: \[ (2x + 13)^2 = 0 \]

4. Take the square root of both sides: \[ 2x + 13 = 0 \]

5. Solve for \( x \): \[ 2x = -13 \] \[ x = -\frac{13}{2} = -6.5 \]

Thus, the solution to the equation \( 4x^2 + 52x + 169 = 0 \) is: \[ \boxed{-6.5} \]