The equation \( x^{2}+5 x-17=0 \) has two solutions \( A \) and \( B \) where \( A

Ask by Sherman Wells. in the United States

Jan 23,2025

To solve the equation \( x^{2}+5x-17=0 \), we can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = 5 \), and \( c = -17 \). Plugging in the values, we get: \[ x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-17)}}{2 \cdot 1} \] Calculating the discriminant: \[ 5^2 - 4 \cdot 1 \cdot (-17) = 25 + 68 = 93 \] Now substituting back into the formula gives us: \[ x = \frac{-5 \pm \sqrt{93}}{2} \] Calculating the two solutions: \[ A = \frac{-5 - \sqrt{93}}{2} \quad \text{and} \quad B = \frac{-5 + \sqrt{93}}{2} \] Now, approximating \( \sqrt{93} \): \[ \sqrt{93} \approx 9.643 \] Thus: \[ A \approx \frac{-5 - 9.643}{2} \approx \frac{-14.643}{2} \approx -7.321 \] \[ B \approx \frac{-5 + 9.643}{2} \approx \frac{4.643}{2} \approx 2.322 \] To three decimal places, we find: \[ A \approx -7.321 \quad \text{and} \quad B \approx 2.322 \] So, \( A = -7.321 \) and \( B = 2.322 \).