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Determining the Nature of the Roots
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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.
People who saw this lesson also found the following lessons useful:
Quadratic Equation Factorization Method
Factorization and Factor Theorem
Rational Expressions - Addition
Elimination by Substitution Method
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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