SOLUTION: How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have differe

Question 471436: How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
To determine if a quadratic will have zero, one, or two (real) solutions, find the discriminant. The discriminant (usually denoted by delta) of a quadratic in the form ax^2 + bx + c is equal to b^2 - 4ac, and the roots of the quadratic are

We see that if the discriminant is zero, then the quadratic will have two roots (equal to -b/2a). If the discriminant is positive the quadratic will have two solutions and if it is negative, it will have no real solutions.

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A well-known theorem says that if r is a root of a polynomial, then x-r is a factor. For example, if a,b are the roots of a polynomial, then (x-a)(x-b) is the polynomial.
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Yes, but only if you scale the quadratic by some constant. For example, (x-a)(x-b) and k(x-a)(x-b) will have the same solutions, if k is a nonzero constant.
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