Questions: Use the quadratic formula to solve the equation 2x^2 - 7x = 1 x = (Simplify your answer. Type an exact answer, using radicals and i as needed. Use integers or fractions

Solution Steps

To solve the quadratic equation \(2x^2 - 7x = 1\), we first rewrite it in the standard form \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions for \(x\).

We start with the given equation: \[ 2x^2 - 7x = 1 \] Rearranging it into standard form gives: \[ 2x^2 - 7x - 1 = 0 \]

From the standard form \(ax^2 + bx + c = 0\), we identify the coefficients:

• \(a = 2\)
• \(b = -7\)
• \(c = -1\)

We calculate the discriminant using the formula \(D = b^2 - 4ac\): \[ D = (-7)^2 - 4 \cdot 2 \cdot (-1) = 49 + 8 = 57 \] Thus, the discriminant is: \[ D = 57 \]

Using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\), we find the solutions: \[ x = \frac{-(-7) \pm \sqrt{57}}{2 \cdot 2} = \frac{7 \pm \sqrt{57}}{4} \]

Calculating the two solutions:

1. For \(x_1\): \[ x_1 = \frac{7 + \sqrt{57}}{4} \approx 3.6375 \]
2. For \(x_2\): \[ x_2 = \frac{7 - \sqrt{57}}{4} \approx -0.1375 \]

Final Answer