Question

solve the quadratic equation.
\\(4x^{2} + 7x - 1 = 0\\)
write one exact, simplified solution in each box. you can add or remove boxes. if a solution is not a real number, write it in the form \\(a + bi\\) or \\(a - bi\\), where \\(a\\) and \\(b\\) are real numbers.

Explanation:

Step1: Identify coefficients

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 4\), \(b = 7\), \(c = -1\).

Step2: Apply quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)
Substitute values: \(x=\frac{-7\pm\sqrt{7^2 - 4(4)(-1)}}{2(4)}\)

Step3: Simplify discriminant

Calculate \(b^2 - 4ac = 49 + 16 = 65\)

Step4: Solve for x

\(x=\frac{-7\pm\sqrt{65}}{8}\)

Answer:

\(\frac{-7 + \sqrt{65}}{8}\), \(\frac{-7 - \sqrt{65}}{8}\)