Question 593740: Write a quadratic equation that has integer coefficients and has as solutions the given pair of numbers 5 and 1
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You want the two roots to occur when X = +5 and when X = +1. Remember what these two roots mean. They mean that for those values of X, the value of the quadratic form f(x) is zero.
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Recall that a quadratic equation can be written in the form:
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ax^2 + bx + c = f(x)
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and you solve for the roots by setting f(x) equal to zero. Then, if you can factor this equation into two factors, you can find the roots by setting each of the factors equal to zero and solving for X.
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For this problem, we are just going in the opposite direction. We know the roots, and we want to get back to the form of the factor that is equal to zero. So let's start with:
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X = +5
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and if we subtract 5 from both sides, this becomes:
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X - 5 = 0
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So X - 5 is one factor.
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Next we can go to
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X = +1
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Subtract 1 from both sides to get:
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X - 1 = 0
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X - 1 is the second factor of this quadratic.
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Multiply the two factors to get the quadratic form:
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(X - 5)(X - 1)
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Multiplying the two factors results in X^2 - 6X + 5
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Therefore, the quadratic equation that has the roots +5 and +1 is:
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f(x) = X^2 - 6X + 5
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and that is the answer to this problem.
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You can check by realizing that if you start with this equation and set f(x) equal to zero to solve for the roots you get:
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X^2 - 6X + 5 = 0
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When you do the factoring you get:
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(X - 5)(X - 1) = 0
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This factored version will be true if either of the factors equals zero. So you get the two real roots by setting X - 5 = 0 and X - 1 = 0 to get the value of the roots as X = +5 and X = +1.
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Maybe working the problem in reverse as we just did will help you make sense of the way that we got from the roots to the corresponding quadratic equation.
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Hope this helps you to understand the problem.
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