Question

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.
$8 x^{2} - 72 = 0$

Accepted Answer

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 $$c = -72. $$Recall the formula for the discriminant: $$\Delta = b^2$ - 4ac. $$This value helps determine the nature and number of solutions of the quadratic equation. Substitute the values of $$a$$, $$b$$, and $$c$$ into the discriminant formula: $$\Delta = 0^2$ - 4 	imes 8 	imes (-72). $$Simplify the expression for the discriminant to find its value (do not calculate the final number, just set up the expression). Use the value of the discriminant to determine the number and type of solutions: if $$\Delta \u003e 0$$, there are two distinct real solutions; if $$\Delta = 0$$, there is one real solution; if $$\Delta \u003c 0$$, there are two nonreal complex solutions. Also, if $$\Delta is a $$perfect square, the solutions are rational; otherwise, they are irrational.

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.
$8 x^{2} - 72 = 0$

Verified video answer for a similar problem:

Key Concepts

Discriminant of a Quadratic Equation

Number and Nature of Solutions Based on the Discriminant

Rational vs. Irrational Solutions

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 9.8x2−72=08x^2 - 72 = 0

Verified video answer for a similar problem:

Key Concepts

Discriminant of a Quadratic Equation

Number and Nature of Solutions Based on the Discriminant

Rational vs. Irrational Solutions

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.
$8 x^{2} - 72 = 0$