Question

solve the equation using the quadratic formula.
10x² = 6x + 3
the solution set is
(type an exact answer, using radicals as needed. use a comma to separate

Explanation:

Step1: Rewrite in standard form

First, we rewrite the equation \(10x^{2}=6x + 3\) in the standard quadratic form \(ax^{2}+bx + c = 0\) by subtracting \(6x\) and \(3\) from both sides:
\(10x^{2}-6x - 3=0\)
Here, \(a = 10\), \(b=-6\), and \(c = - 3\).

Step2: Apply quadratic formula

The quadratic formula is \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
Substitute \(a = 10\), \(b=-6\), and \(c = - 3\) into the formula:
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-6)^{2}-4\times10\times(-3)=36 + 120 = 156\)
Then, \(x=\frac{-(-6)\pm\sqrt{156}}{2\times10}=\frac{6\pm\sqrt{4\times39}}{20}=\frac{6\pm2\sqrt{39}}{20}\)
Simplify the fraction by dividing numerator and denominator by 2:
\(x=\frac{3\pm\sqrt{39}}{10}\)

Answer:

\(\frac{3 + \sqrt{39}}{10},\frac{3 - \sqrt{39}}{10}\)