To solve the quadratic equation \(4x^2 + 52x + 169 = 0\) by factoring, we need to check if it can be expressed as a perfect square trinomial. A perfect square trinomial has the form \((ax + b)^2 = a^2x^2 + 2abx + b^2\).
Let's start by rewriting the equation:
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Factor out the coefficient of \(x^2\) (which is 4 in this case):
\[ 4(x^2 + 13x) + 169 = 0 \]
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Now, rearrange the equation:
\[ 4(x^2 + 13x + \text{something}) + 169 - \text{something} = 0 \]
We can complete the square for the expression inside the parentheses. To do this, we take half of the coefficient of \(x\) (which is 13), square it, and add it inside:
Half of 13 is \( \frac{13}{2} \) and squaring it gives:
\[ \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
Now we can express this trinomial as a perfect square:
\[ x^2 + 13x + \frac{169}{4} = \left(x + \frac{13}{2}\right)^2 \]
Therefore, we rewrite our original equation after completing the square:
\[ 4\left(x + \frac{13}{2}\right)^2 - 4 \cdot \frac{169}{4} + 169 = 0 \]
Simplifying gives:
\[ 4\left(x + \frac{13}{2}\right)^2 = 0 \]
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Set the equation to zero:
\[ \left(x + \frac{13}{2}\right)^2 = 0 \]
Taking the square root of both sides, we find:
\[ x + \frac{13}{2} = 0 \]
So,
\[ x = -\frac{13}{2} = -6.5 \]
The solution to the quadratic equation \(4x^2 + 52x + 169 = 0\) is \(x = -6.5\).
Answer: \(x = -6.5\)