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 Depdendent Variable

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# The Discriminant

In the quadratic formula, the radicand, b2 - 4ac, is called the discriminant of the quadratic equation ax2 + bx + c = 0.

We can use the discriminant to determine the nature of the solutions of a quadratic equation without having to solve the equation.

 Discriminant Solutions b2 - 4ac > 0 two different real numbers b2 - 4ac = 0 two identical real numbers b2 - 4ac < 0 no real number solutions

If the discriminant is a perfect square, the solutions will not only be real numbers, they will also be rational numbers.

Example

Use the discriminant to determine the nature of the solutions of this quadratic equation: -2x2 - x + 7 = 0

Solution

 The equation has the form ax2 + bx + c = 0 where a = -2, b = -1, and c = 7. Substitute the values of a, b, and c into  the discriminant and simplify. b2 - 4ac = (-1)2 - 4(-2)(7) = 1 + 56 = 57
The discriminant is 57, a positive number.

So the equation -2x2 - x + 7 = 0 has two unequal real number solutions.