Question 535981: One solution of kx^2-5x+k=0 is 3. Find the other solution
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Problem: if one solution of the quadratic equation:
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is 3, what is the other solution?
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Notice that when you term-by-term compare the given equation to the standard form of the quadratic equation:
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you can see that "a" correlates to k, "b" equals -5, and "c" also equals k.
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We know that the quadratic formula applies to solving a quadratic equation in the standard form. The quadratic formula says that for a quadratic equation in the standard form x can be found from:
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into this formula we can substitute k for "a" and "c" and -5 for "b" to get:
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The  becomes + 5 and the  and  . These actions result in:
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multiply both sides by 2*k which is the denominator on the right side. This clears the denominator, and the formula becomes:
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we also know that one value for x is 3. We were told that in the statement of the problem. So substitute 3 for x in the formula. This makes the left side become:
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multiply out the left side to get:
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get rid of the 5 on the right side by subtracting 5 from both sides:
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square both sides to get rid of the radical on the right side and this formula becomes:
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cancel 25 on both sides of the formula by subtracting 25 from both sides. Also get rid of the  on the right side by adding  to both sides to get:
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Factor a k from terms on the left side:
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notice that this equation will be true if either of the two factors on the left side is equal to zero because multiplication by a zero results in the left side becoming zero and making the left side equal to the zero on the right side.
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So this formula will be true if either k = 0 or if 40k - 60 = 0. We can ignore k = 0 because if this were true in the original quadratic equation of the problem if k were zero, both the 
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then dividing both sides by 60:
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and reducing the right side by dividing both the numerator and denominator by 20 to get:
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Now we can go back to the original problem and substitute  for k to make it become:
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Multiply all terms by 2 to cancel out the denominator and you have:
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compared to the standard quadratic equation we now have "a" = 3, "b" = -10, and "c" = 3. Substituting these values into the quadratic formula, we get:
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which becomes:
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Simplifying to:
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The radical term becomes the square root of 64:
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which simplifies to:
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 and this simplifies to 
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and
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 and this simplifies to 
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The first value for x we already knew because it was part of the problem. The second value for x, which we found was , is the answer you were asked to find in the problem.
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Hope this helps you to understand the quadratic equation a little better and how to apply that knowledge to solving this problem.
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Check my work. It's late and I might have introduced some errors due to lack of coffee. You can also check this problem by multiplying the two factors:
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 and see if the answer does not give you the original equation in the problem in which
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Good luck ...
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