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.
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Rewrite the quadratic: We can express the quadratic in the standard form \(ax^2 + bx + c\), where \(a = 4\), \(b = 52\), and \(c = 169\).
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Identify the perfect square: We can factor out the coefficient of \(x^2\): \[ 4(x^2 + 13x) + 169 = 0 \]
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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: \[ 4\left(x + \frac{13}{2}\right)^2 = 0 \]
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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
\[ \boxed{-\frac{13}{2}}. \]
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