Question

use the discriminant to determine the number and types of solutions of the quadratic equation.
-2x = -3x² + 6
the equation has
two real solutions
two complex but not real solutions
one real solution

Explanation:

Step1: Rewrite in standard form

First, rewrite the quadratic equation \(-2x = -3x^2 + 6\) in standard form \(ax^2 + bx + c = 0\). Add \(3x^2\) to both sides and subtract \(6\) from both sides:
\(3x^2 - 2x - 6 = 0\). Here, \(a = 3\), \(b = -2\), \(c = -6\).

Step2: Calculate the discriminant

The discriminant of a quadratic \(ax^2 + bx + c\) is \(D = b^2 - 4ac\). Substitute \(a = 3\), \(b = -2\), \(c = -6\):
\(D = (-2)^2 - 4(3)(-6)\)
\(D = 4 + 72\)
\(D = 76\).

Step3: Analyze the discriminant

Since \(D = 76 > 0\), a positive discriminant means the quadratic has two distinct real solutions.

Answer:

two real solutions