Evaluate the discriminant for each equation. Then use it to determine the number of distinct solutions, and tell whether they are rational, irrational, or nonreal complex numbers. (Do not solve the equation.) 4x² = -6x + 3

Verified step by step guidance

1. Rewrite the given equation in standard quadratic form \(ax^2 + bx + c = 0\). Start by moving all terms to one side: \(4x^2 + 6x - 3 = 0\).
2. Identify the coefficients: \(a = 4\), \(b = 6\), and \(c = -3\) from the standard form.
3. Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\).
4. Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = (6)^2 - 4 \times 4 \times (-3)\).
5. Analyze the discriminant value to determine the nature of the roots: if \(\Delta > 0\) and a perfect square, roots are rational; if \(\Delta > 0\) but not a perfect square, roots are irrational; if \(\Delta = 0\), one real rational root; if \(\Delta < 0\), roots are nonreal complex numbers.

Key Concepts

Discriminant of a Quadratic Equation

The discriminant is a value calculated from the coefficients of a quadratic equation in the form ax² + bx + c = 0, given by b² - 4ac. It helps determine the nature and number of solutions without solving the equation.

Number and Nature of Solutions Based on the Discriminant

If the discriminant is positive, there are two distinct real solutions; if zero, one real repeated solution; if negative, two nonreal complex solutions. This classification guides understanding the roots' behavior.

Rational vs. Irrational Solutions

When the discriminant is a perfect square, the solutions are rational numbers; if it is positive but not a perfect square, the solutions are irrational. This distinction helps describe the exact type of real roots.