FACTORIZE \( 4 x^{2}-20 x-9 \)

Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(4x^{2}-20x-9=0\) - step1: Solve using the quadratic formula: \(x=\frac{20\pm \sqrt{\left(-20\right)^{2}-4\times 4\left(-9\right)}}{2\times 4}\) - step2: Simplify the expression: \(x=\frac{20\pm \sqrt{\left(-20\right)^{2}-4\times 4\left(-9\right)}}{8}\) - step3: Simplify the expression: \(x=\frac{20\pm \sqrt{544}}{8}\) - step4: Simplify the expression: \(x=\frac{20\pm 4\sqrt{34}}{8}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{20+4\sqrt{34}}{8}\\&x=\frac{20-4\sqrt{34}}{8}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{5+\sqrt{34}}{2}\\&x=\frac{20-4\sqrt{34}}{8}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{5+\sqrt{34}}{2}\\&x=\frac{5-\sqrt{34}}{2}\end{align}\) - step8: Rewrite: \(x_{1}=\frac{5-\sqrt{34}}{2},x_{2}=\frac{5+\sqrt{34}}{2}\) To factor the expression \( 4x^{2} - 20x - 9 \), we first need to find its roots by solving the equation \( 4x^{2} - 20x - 9 = 0 \). ### Step 1: Solve the Quadratic Equation Using the quadratic formula, we find the roots: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 4 \), \( b = -20 \), and \( c = -9 \). Calculating the discriminant: \[ b^2 - 4ac = (-20)^2 - 4 \cdot 4 \cdot (-9) = 400 + 144 = 544 \] Now, substituting into the quadratic formula: \[ x = \frac{20 \pm \sqrt{544}}{8} \] ### Step 2: Simplify the Roots The square root of 544 can be simplified: \[ \sqrt{544} = \sqrt{16 \cdot 34} = 4\sqrt{34} \] Thus, the roots are: \[ x_{1} = \frac{20 - 4\sqrt{34}}{8} = \frac{5 - \sqrt{34}}{2} \] \[ x_{2} = \frac{20 + 4\sqrt{34}}{8} = \frac{5 + \sqrt{34}}{2} \] ### Step 3: Write the Factored Form Using the roots, we can express the quadratic in its factored form: \[ 4x^{2} - 20x - 9 = 4\left(x - \frac{5 - \sqrt{34}}{2}\right)\left(x - \frac{5 + \sqrt{34}}{2}\right) \] ### Final Factored Form Thus, the factored form of \( 4x^{2} - 20x - 9 \) is: \[ 4\left(x - \frac{5 - \sqrt{34}}{2}\right)\left(x - \frac{5 + \sqrt{34}}{2}\right) \]