SOLUTION: I am attempting to write an quadratric equation that has the solutions 10 and 2 and I am coming up empty. I could use some help please.

Question 533824: I am attempting to write an quadratric equation that has the solutions 10 and 2 and I am coming up empty. I could use some help please.
Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
Try this.
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Assume that the quadratic expression has two factors:
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(x + A)*(x + B)
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The roots of such the equation are the values of x that make the expression equal to zero. In this case we are told that when x is +10 or +2 the quadratic expression will equal zero. Substitute these values for x in the two factors:
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(10 + A)*(2 + B)
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The product of these two factors will equal zero if either of the two factors is equal to zero. To find the values of A and B that will make the factors equal to zero, set each factor equal to zero:
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10 + A = 0
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Solve for A by subtracting 10 from both sides:
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A = -10
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Similarly set the second factor equal to zero:
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2 + B = 0
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Solve for B by subtracting 2 from both sides:
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B = -2
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Now write the two factors substituting these values for A and B:
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(x + A)*(x + B) = (x - 10)*(x - 2)
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Notice on the right side where A and B values have been substituted that if x = 10 or if x = 2 (the required roots) one of the factors will become zero. Let's multiply the two factors:
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(x - 10)*(x - 2) = x^2 - 2x - 10x + 20
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Combine the two terms that contain x and the resulting expression is:
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x^2 - 12x + 20
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Set this quadratic expression equal to y to establish the quadratic equation:
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x^2 - 12x + 20 = y
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Notice that if we substitute x = 10 the right side of the equation (y) becomes zero:
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100 - 120 + 20 = 0
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Therefore, x = 10 is a root (x-axis crossing point) of this equation.
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Next substitute 2 for x and the expression becomes:
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4 - 24 + 20 = 0
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So x = 2 is also a root (another x-axis crossing) of the equation.
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This means that the quadratic equation of:
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x^2 - 12x + 20 = y
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Is the answer to this problem because it has the values x = 10 and x = 2 as roots of the equation.
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Now after all this complex explanation, here's the short way to remember how to do problems like this. Whenever you are given the desired roots, just change their sign and add them with x to form two factors, then multiply the two factors together and set the product equal to y and you have the quadratic equation.
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Here's an example to show you how fast it can go. Suppose you want to find a quadratic equation having roots of -4 and +3. Write the two factors as a product and set it equal to y to get:
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(x + 4)*(x - 3) = y
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Note that in the factors the desired roots have the opposite sign.
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Multiply out the left side and you have:
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x^2 + x - 12 = y
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And that's the answer. You can substitute -4 for x and then +3 for x and you will find that in either case the substitution makes y equal to zero. Therefore, the roots of the equation are -4 and +3, just as we wanted them to be.
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Hope this helps you in understanding this problem.
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