# Using and Understanding the Discriminant – Practice ProblemsHaving fun while studying, practice your skills by solving these exercises!

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A quadratic equation in standard form ax² + bx + c = 0 can be solved in various ways. These include factoring, completing the square, graphing, and using the quadratic formula. Solving for the roots or solutions of a quadratic equation by using the quadratic formula is often the most convenient way. The solutions to a quadratic equation can be determined by this quadratic formula: x = [-b ± √(b²-4ac)] / 2a. We can predict the number of solutions to a quadratic equation by evaluating the discriminant given by b²-4ac or the radicand expression in the quadratic formula. The value of the discriminant determines the nature and number of solutions of a quadratic equation.

If b²-4ac is positive, there will be two distinct, real solutions.

For example, the quadratic equation x² - 7x + 12 = 0 has two distinct, real solutions since b²-4ac = (-7)² - 4(1)(12) = 49 - 48 = 1 is positive.

If b²-4ac is zero, there will be one distinct, real solution.

For example, the quadratic equation 4x² - 4x + 1 = 0 has one distinct, real solution since b²-4ac = (-4)² - 4(4)(1) = 16 - 16 = 0.

If b2-4ac is negative, there are no real solutions.

For example, the quadratic equation 5x² + 2x + 3 = 0 has no real solutions since b2-4ac = (2)² - 4(5)(3) = 4 - 60 = -56 is negative.

Solve quadratic equations with real coefficients that have complex solutions.

CCSS.MATH.CONTENT.HSN.CN.C.7

Exercises in this Practice Problem
 Define the discriminant. Determine the number of solutions. Find the discriminants. Determine which equation has two solutions. Explain what a discriminant equal to $100$ tells you. Calculate the discriminant.