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.) See Example 9.
$8x^2-72=0$

Verified step by step guidance

1. Identify the coefficients of the quadratic equation in the standard form \(ax^2 + bx + c = 0\). For the equation \$8x^2 - 72 = 0\(, note that \)a = 8\(, \)b = 0\(, and \)c = -72$.
2. Recall the formula for the discriminant: \(\Delta = b^2 - 4ac\). This value helps determine the nature and number of solutions of the quadratic equation.
3. Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(\Delta = 0^2 - 4 \times 8 \times (-72)\).
4. Simplify the expression for the discriminant to find its value (do not calculate the final number, just set up the expression).
5. Use the value of the discriminant to determine the number and type of solutions: if \(\Delta > 0\), there are two distinct real solutions; if \(\Delta = 0\), there is one real solution; if \(\Delta < 0\), there are two nonreal complex solutions. Also, if \(\Delta\) is a perfect square, the solutions are rational; otherwise, they are irrational.

Key Concepts

Discriminant of a Quadratic Equation

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

Number and Nature of Solutions Based on the Discriminant

The discriminant indicates the number and type of roots: if Δ > 0, there are two distinct real solutions; if Δ = 0, there is one real repeated solution; if Δ < 0, there are two nonreal complex solutions.

Rational vs. Irrational Solutions

When the discriminant is a perfect square and positive, the solutions are rational numbers. If the discriminant is positive but not a perfect square, the solutions are irrational. This distinction helps classify the exact nature of the roots.