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 Depdendent Variable

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 Dependent Variable

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# The Discriminant

Example 1

Use the discriminant to determine the nature of the solutions of this quadratic equation: 5x2 - 3x + 8 = 0

Solution

The equation has the form ax2 + bx + c = 0 where a = 5, b= -3, and c = 8.

 The equation has the form ax2 + bx + c = 0 where a = 5, b = -3, and c = 8. Substitute the values of a, b, and c into  the discriminant and simplify. b2 - 4ac = (-3)2 - 4(5)(8) = 9 - 160 = -151
The discriminant is -151, a negative number.

So the equation 5x2 - 3x + 8 = 0 has no real number solutions.

Example 2

Use the discriminant to determine the nature of the solutions of this quadratic equation: 9x2 - 6x = -1

 Solution To put the equation in standard form, add 1 to both sides of the equation. Now the equation has the form ax2 + bx + c = 0 where a = 9, b = - 6, and c = 1. 9x2 - 6x = -1 9x2 - 6x + 1 = 0 Substitute the values of a, b, and c into the discriminant and simplify. b2 - 4ac = (-6)2 - 4(9)(1)= 36 - 36 = 0

The discriminant is 0.

So the equation 9x2 - 6x = -1 has two identical real number solutions.