Question

solve the following equation by either factoring or quadratic formula $5x^2 - 13x = 6$
enter your answer as a decimal
$x_1 = \square$
$x_2 = \square$

Explanation:

Step1: Rewrite the equation

First, rewrite the equation \(5x^{2}-13x = 6\) in standard quadratic form \(ax^{2}+bx + c=0\). Subtract 6 from both sides:
\(5x^{2}-13x - 6=0\)
Here, \(a = 5\), \(b=- 13\), \(c=-6\).

Step2: Use the quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac\).
Substitute \(a = 5\), \(b=-13\), \(c = - 6\) into the discriminant formula:
\(\Delta=(-13)^{2}-4\times5\times(-6)=169 + 120=289\)
Then, find the square root of the discriminant: \(\sqrt{\Delta}=\sqrt{289} = 17\)
Now, substitute \(a\), \(b\) and \(\sqrt{\Delta}\) into the quadratic formula:
\(x=\frac{-(-13)\pm17}{2\times5}=\frac{13\pm17}{10}\)

Step3: Find the two solutions

For the plus sign:
\(x_{1}=\frac{13 + 17}{10}=\frac{30}{10}=3\)
For the minus sign:
\(x_{2}=\frac{13-17}{10}=\frac{-4}{10}=- 0.4\)

Answer:

\(x_1 = 3\)
\(x_2=-0.4\)