Using and Understanding the Discriminant – Practice Problems

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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

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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.