To solve the quadratic equation \( 4x^2 + 52x + 169 = 0 \) by factoring using the perfect square trinomial pattern, we can start by rewriting the left-hand side:
- We first notice that the quadratic can be rewritten in the form of \((ax + b)^2\).
The general form for a perfect square trinomial is: \[ (ax + b)^2 = a^2x^2 + 2abx + b^2 \]
In our equation, we identify:
- \( a^2 = 4 \) which means \( a = 2 \),
- \( b^2 = 169 \) which gives \( b = 13 \).
Now, we need to check if the middle term \( 2ab \) corresponds to the middle term in our equation: \[ 2ab = 2 \cdot 2 \cdot 13 = 52 \] This matches the coefficient of \( x \) in our equation, so we can indeed factor the quadratic.
Now we can write: \[ 4x^2 + 52x + 169 = (2x + 13)^2 \]
Now, we set the equation equal to zero: \[ (2x + 13)^2 = 0 \]
To solve for \( x \), we take the square root of both sides: \[ 2x + 13 = 0 \]
Subtract 13 from both sides: \[ 2x = -13 \]
Now divide by 2: \[ x = -\frac{13}{2} = -6.5 \]
Thus, the solution to the equation \( 4x^2 + 52x + 169 = 0 \) is: \[ \boxed{-6.5} \]
Checking the options provided:
- \(x = -8.45\)
- \(x = -6.5\) (this is correct)
- \(x = 8.45\)
- \(x = 6.5\)
The correct solution from the options is \(x = -6.5\).
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