Questions: Solve the quadratic equation by completing the square. x^2 + 8x = 48 The solution set is .

Solution Steps

Start with the original equation: \[ x^{2} + 8x = 48 \] Rearranging gives: \[ x^{2} + 8x - 48 = 0 \]

To complete the square, take half of the coefficient of \( x \) (which is \( 8 \)), square it, and add it to both sides. Half of \( 8 \) is \( 4 \), and squaring it gives \( 16 \): \[ x^{2} + 8x + 16 = 48 + 16 \] This simplifies to: \[ (x + 4)^{2} = 64 \]

Now, take the square root of both sides: \[ x + 4 = \pm \sqrt{64} \] This leads to: \[ x + 4 = \pm 8 \] Isolating \( x \) gives two solutions: \[ x = -4 + 8 \quad \text{and} \quad x = -4 - 8 \] Thus, the solutions are: \[ x = 4 \quad \text{and} \quad x = -12 \]

Final Answer