\( \left. \begin{array} { | l | } \hline \text { Use the quadratic formula to solve the equations. } \\ \hline 37 . - 7 x ^ { 2 } - 5 x + 1 = 0 \\ \hline \end{array} \right. \)

To solve the quadratic equation \( -7x^2 - 5x + 1 = 0 \) using the quadratic formula, we first need to identify the coefficients \( a \), \( b \), and \( c \) from the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). From the equation \( -7x^2 - 5x + 1 = 0 \): - \( a = -7 \) - \( b = -5 \) - \( c = 1 \) The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Now, let's calculate the discriminant \( b^2 - 4ac \): 1. Calculate \( b^2 \): \[ b^2 = (-5)^2 = 25 \] 2. Calculate \( 4ac \): \[ 4ac = 4 \cdot (-7) \cdot 1 = -28 \] 3. Now, calculate the discriminant: \[ b^2 - 4ac = 25 - (-28) = 25 + 28 = 53 \] Now we can substitute \( a \), \( b \), and the discriminant into the quadratic formula: \[ x = \frac{-(-5) \pm \sqrt{53}}{2 \cdot (-7)} \] This simplifies to: \[ x = \frac{5 \pm \sqrt{53}}{-14} \] Now, we can express the two solutions: 1. \( x_1 = \frac{5 + \sqrt{53}}{-14} \) 2. \( x_2 = \frac{5 - \sqrt{53}}{-14} \) Let's calculate these values. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(5+\sqrt{53}\right)}{\left(-14\right)}\) - step1: Remove the parentheses: \(\frac{5+\sqrt{53}}{-14}\) - step2: Rewrite the fraction: \(-\frac{5+\sqrt{53}}{14}\) Calculate or simplify the expression \( (5 - \sqrt(53)) / -14 \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(5-\sqrt{53}\right)}{\left(-14\right)}\) - step1: Remove the parentheses: \(\frac{5-\sqrt{53}}{-14}\) - step2: Rewrite the fraction: \(-\frac{5-\sqrt{53}}{14}\) - step3: Alternative Form: \(\approx 0.162865\) The solutions to the equation \( -7x^2 - 5x + 1 = 0 \) using the quadratic formula are: 1. \( x_1 = \frac{5 + \sqrt{53}}{-14} \approx -0.877 \) 2. \( x_2 = \frac{5 - \sqrt{53}}{-14} \approx 0.163 \) Thus, the final answers are: - \( x_1 \approx -0.877 \) - \( x_2 \approx 0.163 \)