To solve the quadratic equation \(4x^2 + 52x + 169 = 0\) by factoring using the perfect square trinomial pattern, we first need to express the left-hand side as a perfect square.
The general form of a perfect square trinomial is \((ax + b)^2 = a^2x^2 + 2abx + b^2\).
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Factor out the coefficient of \(x^2\): \[ 4x^2 + 52x + 169 = 4(x^2 + 13x) + 169 \]
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Complete the square: To complete the square for the expression \(x^2 + 13x\), we take half of the coefficient of \(x\) (which is \(13\)), square it, and add and subtract that square inside the parentheses.
Half of \(13\) is \(\frac{13}{2}\), and squaring it gives: \[ \left(\frac{13}{2}\right)^2 = \frac{169}{4} \]
Now, we add and subtract \(\frac{169}{4}\): \[ x^2 + 13x = \left(x^2 + 13x + \frac{169}{4} - \frac{169}{4}\right) = \left(x + \frac{13}{2}\right)^2 - \frac{169}{4} \]
Therefore: \[ 4\left(x^2 + 13x\right) = 4\left(\left(x + \frac{13}{2}\right)^2 - \frac{169}{4}\right) = 4\left(x + \frac{13}{2}\right)^2 - 169 \]
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Rewrite the original equation: Substituting back we have: \[ 4\left(x + \frac{13}{2}\right)^2 - 169 + 169 = 0 \] Simplifying gives: \[ 4\left(x + \frac{13}{2}\right)^2 = 0 \]
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Solving for \(x\): Dividing both sides by \(4\): \[ \left(x + \frac{13}{2}\right)^2 = 0 \] Taking the square root of both sides: \[ x + \frac{13}{2} = 0 \] Therefore: \[ x = -\frac{13}{2} \]
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Converting to decimal: Dividing \(-13\) by \(2\): \[ x = -6.5 \]
So the solution to the equation \(4x^2 + 52x + 169 = 0\) is: \[ \boxed{-6.5} \]
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