Questions: Solve by completing the square. x^2 + (13/2) x = 13 The solution(s) is/are x=

Solution Steps

To solve the quadratic equation by completing the square, follow these steps:

1. Move the constant term to the right side of the equation.
2. Divide the coefficient of the linear term by 2 and square it.
3. Add this square to both sides of the equation to form a perfect square trinomial on the left side.
4. Factor the perfect square trinomial.
5. Solve for \( x \) by taking the square root of both sides and isolating \( x \).

We start with the equation: \[ x^{2} + \frac{13}{2} x = 13 \] We rearrange it to isolate the constant on one side: \[ x^{2} + \frac{13}{2} x - 13 = 0 \]

To complete the square, we take the coefficient of \( x \), which is \( \frac{13}{2} \), divide it by 2, and square it: \[ \left(\frac{13}{4}\right)^{2} = \frac{169}{16} \] We add this value to both sides of the equation: \[ x^{2} + \frac{13}{2} x + \frac{169}{16} = 13 + \frac{169}{16} \]

To simplify the right side, we convert 13 to a fraction with a denominator of 16: \[ 13 = \frac{208}{16} \] Thus, we have: \[ x^{2} + \frac{13}{2} x + \frac{169}{16} = \frac{208}{16} + \frac{169}{16} \] This simplifies to: \[ x^{2} + \frac{13}{2} x + \frac{169}{16} = \frac{377}{16} \]

The left side can be factored as: \[ \left(x + \frac{13}{4}\right)^{2} = \frac{377}{16} \]

Taking the square root of both sides gives: \[ x + \frac{13}{4} = \pm \sqrt{\frac{377}{16}} \] This simplifies to: \[ x + \frac{13}{4} = \pm \frac{\sqrt{377}}{4} \] Isolating \( x \) results in: \[ x = -\frac{13}{4} \pm \frac{\sqrt{377}}{4} \] Thus, the solutions are: \[ x = \frac{-13 \pm \sqrt{377}}{4} \]

Final Answer