SOLUTION: If z=3 is a solution of the equation 5z^2+kz-30=0 the value of k and other solution of the equation are respectively -5 and 2 -5 and -2 5 and 2

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> SOLUTION: If z=3 is a solution of the equation 5z^2+kz-30=0 the value of k and other solution of the equation are respectively -5 and 2 -5 and -2 5 and 2      Log On


   

Question 1136528: If z=3 is a solution of the equation 5z^2+kz-30=0 the value of k and other solution of the equation are respectively
-5 and 2
-5 and -2
5 and 2
Found 3 solutions by josmiceli, greenestamps, ikleyn:
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
+z+=+3+
+5z%5E2+%2B+k%2Az+-+30+=+0+
+5%2A3%5E2+%2B+k%2A3+-+30+=+0+
+45+%2B+3k+=+30+
+3k+=+-15+
+k+=+-5+
-------------------
+5z%5E2+-+5z+-+30+=+0+
+z%5E2+-+z+-+6+=+0+
+%28+z+-+3+%29%2A%28+z+%2B+2+%29++=+0+
--------------------------------
+z+=+3+
+z+=+-2+


Answer by greenestamps(13203) About Me  (Show Source):
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z=3 is a solution to the quadratic equation, so (z-3) is a factor of the quadratic expression. Find the other factor.

%28z-3%29%28az%2Bb%29+=+5z%5E2%2Bkz-30

For the leading coefficient of the quadratic to be 5, a must be 5.
For the constant term in the quadratic to be -30, b has to be 10.

%28z-3%29%285z%2B10%29+=+5z%5E2-5z-30+=+5z%5E2%2Bkz-30

k is -5; and the second root is -2

Answer by ikleyn(52855) About Me  (Show Source):
You can put this solution on YOUR website!
.
The given equation is equivalent to quadratic equation

    z%5E2+%2B+%28k%2F5%29%2Az+-+6 = 0     (1)

with the leading coefficient 1.  (Notice that equation (1) is obtained from the given equation by division all the terms by 5.)


Therefore, equation (1) has the root z= 3.


According to Vieta's theorem, the product of the roots of the equation (1) is equal to its constant term, which is -6.

Therefore, the second root of the equation (1) is  -6%2F3 = -2.


Thus the two roots of the equation (1) are 3 and -2.


Then, according to the Vieta's theorem, the sum of its roots (which is  3+(-2) = 1) is equal to the coefficient at x with the opposite sign.


In other words,  -k%2F5 = 3+(-2) = 1,  which implies  k = -5.


ANSWER.  The second root of the given equation is -2 and k= -5.

Solved.