Question

using the quadratic formula, which (select all) are solutions to:

$5x^2 - 11x + 4 = 0$

(make sure that you understand from the video what it means if asked to show all work)

• $-11$
• $\frac{11+sqrt{41}}{10}$
• $4$
• $\frac{11-sqrt{41}}{10}$

Explanation:

Step1: Recall quadratic formula

The quadratic formula for a quadratic equation \(ax^{2}+bx + c = 0\) is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\). For the equation \(5x^{2}-11x + 4=0\), we have \(a = 5\), \(b=- 11\), \(c = 4\).

Step2: Calculate discriminant

First, calculate the discriminant \(D=b^{2}-4ac\). Substitute \(a = 5\), \(b=-11\), \(c = 4\) into the formula:
\(D=(-11)^{2}-4\times5\times4=121 - 80=41\)

Step3: Apply quadratic formula

Now, substitute \(a = 5\), \(b=-11\), \(D = 41\) into the quadratic formula:
\(x=\frac{-(-11)\pm\sqrt{41}}{2\times5}=\frac{11\pm\sqrt{41}}{10}\)
So the solutions are \(\frac{11 + \sqrt{41}}{10}\) and \(\frac{11-\sqrt{41}}{10}\)

Answer:

\(\boldsymbol{\frac{11+\sqrt{41}}{10}}\), \(\boldsymbol{\frac{11-\sqrt{41}}{10}}\) (corresponding to the options \(\frac{11+\sqrt{41}}{10}\) and \(\frac{11-\sqrt{41}}{10}\))