Determining the Nature of the Roots

 Determining the Nature of the Roots If you have a quadratic equation ax2 + bx + c = 0 , then it is true that x =[-b (b2 - 4ac)]/2a. That (b2 - 4ac) is known as the discriminant (). If > 0, then the roots are real and distinct. Further, if > 0 and is a perfect square, then the roots are rational. If = 0, there is one solution to the quadratic equation, or two equal roots. If < 0, then the roots not real, incorporating an imaginary component.
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 Example: If -5 is a root of the quadratic equation 2x2 + px - 15 = 0 and the quadratic equation p(x2 + x) + k = 0 has equal roots, find the value of k. Solution: Since -5 is a root of the equation 2x2 + px - 15 = 0. Therefore, 2(-5)2 - 5p - 15 - 0 = 0 => 50 - 5p - 15 = 0 => 5p = 35 => p = 7. Putting p = 7 in p(x2 + x) + k = 0, we get 7x2 + 7x + k = 0 This equation will have equal roots, if discriminant = 0 => 49 - 4 x 7 x k = 0 => k = 49/28 => k = 7/4

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