Question

use the quadratic formula to solve the equation. the equation has real number solutions.\\(\frac{1}{4}x^2 - 5x - 4 = 0\\)\\(x = \square\\)\\(\text{(simplify your answer, using radicals as needed. do not use decimal approximations. use a comma}\\)

Explanation:

Step1: Identify a, b, c values

For $\frac{1}{4}x^2 - 5x - 4 = 0$, $a=\frac{1}{4}$, $b=-5$, $c=-4$

Step2: Recall quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step3: Compute discriminant

$\Delta = b^2-4ac = (-5)^2 - 4\times\frac{1}{4}\times(-4) = 25 + 4 = 29$

Step4: Substitute into formula

$x=\frac{-(-5)\pm\sqrt{29}}{2\times\frac{1}{4}} = \frac{5\pm\sqrt{29}}{\frac{1}{2}}$

Step5: Simplify the expression

$x=(5\pm\sqrt{29})\times2 = 10\pm2\sqrt{29}$

Answer:

$10 + 2\sqrt{29}, 10 - 2\sqrt{29}$