Question

solve the following equation using the quadratic formula.
3x² + 2x - 5 = 0
the solution set is
(type an exact answer, using radicals and i as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.)

Explanation:

Step1: Identify coefficients

For equation \(3x^2 + 2x - 5 = 0\), \(a = 3\), \(b = 2\), \(c = -5\).

Step2: Quadratic formula

Quadratic formula: \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Calculate discriminant: \(b^2 - 4ac=(2)^2 - 4(3)(-5)=4 + 60 = 64\).

Step3: Substitute values

Substitute \(a\), \(b\), discriminant into formula:
\(x=\frac{-2\pm\sqrt{64}}{2(3)}=\frac{-2\pm8}{6}\).

Step4: Find solutions

First solution: \(\frac{-2 + 8}{6}=\frac{6}{6}=1\).
Second solution: \(\frac{-2 - 8}{6}=\frac{-10}{6}=-\frac{5}{3}\).

Answer:

\(1, -\frac{5}{3}\)