Found 3 solutions by ikleyn, lotusjayden, MathTherapy:
Answer by ikleyn(52813) (Show Source): You can put this solution on YOUR website!
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Solve these two equations:
(a) The quadratic equation has two roots.
If one root is five times the other, what is the value of a?
(b) If and are factors of the polynomial ,
find .
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Part (a)
Let z be one root of the equation; then 5z is another root.
Apply the Vieta's theorem. it gives you these two equations for unknowns "a" and "z":
(1) the sum of the roots z and 5z is equal to = 2
z + 5z = 2, or 6z = 2, z = = .
(2) the product of the roots is equal to
z*(5z) = , which implies = ,
= , 5*(2a) = 9*(a+1), 10a = 9a + 9, a = 9.
Thus we get the ANSWER : a = 9.
Part (a) is solved.
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Two problems in your post are different.
At this forum, I do not recommend to pack more than one problem to post, in order for do not create a MESS.
Therefore, post your second problem separately, and I will show you how to solve it.
By the way, the rules of this forum PROHIBIT posting more than one problem at once.
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Answer by lotusjayden(18) (Show Source): You can put this solution on YOUR website!
I'll solve this part b by myself.
(b) If and are factors of the polynomial ,
find .
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First, you should multiply the two known factors together:
which simplifies to:
after you get that, one method to solve this problem is to do trial and error:
After a while, you should get:
The simplified version is:
Now, you can see that , and .
Thus we get the ANSWER : a = .
Solved.
Answer by MathTherapy(10555) (Show Source): You can put this solution on YOUR website!
Solve these two equations:
(a) The quadratic equation has two roots. If one root is five times the other, what is the value of a?
(b) If and are factors of the polynomial , find .
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Let smaller root be R
Then larger root = 5R
Sum of the roots, or . Also, sum of roots = R + 5R = 6R.
Therefore,
Product of the roots, or . Also, product of roots = R(5R) = 5R2 =
Therefore,
10a = 9(a + 1)----- Cross-multiplying
10a = 9a + 9
10a - 9a = 9
***BTW, this is Part (a).
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