Question

Find the value of the the constant k for which the equation gf(x)= k has equal roots.

f:x |--> 4x-2x^2
g:x |--> 5x+3

Answer 1

is it g(f(x)) = k?

Answer 2

No. It's gf(x).
Which is
5(4x-2x^2)+3

Answer 3

Using the standard formula for finding the roots of a quadratic and substituting with C being the required constant for equal roots the following term must equal zero.

\[\sqrt{(400}+40C)=0\]
Therefore the required constant C=-10

Answer 4

I thought you used b-4ac? And that's not the answer...?

Answer 5

But how do you plug in the values into b-4ac?

Answer 6

The roots are 1,1

Answer 7

i got k=13. do you know the answer? i think if finally understand the question..

Answer 8

I agree with dpalnc. I forgot to do the last step:(

Answer 9

I used:
\[b ^{2}-4ac\]

Answer 10

How? I don't know how to plug in the values..?

Answer 11

I first rearranged as follows:
\[-10x ^{2}+20x+C\]
Then a=-10, b=20 and c=C
In that case:
\[b ^{2}-4ac=400-4(-10*C)=0\] when C=-10
Therefore the required constant on the RHS of the original equation must be 13. Subtracting 13 from both sides makes the quadratic equal to zero and the constant term becomes -10.

Answer 12

That doesn't seem to make much sense. C is already given... =3, and the workings don't match..

Answer 13

I ignored the given constant 3 and concentrated on finding what constant is required to form a quadratic equation using the two given variable terms. Having found what constant was needed it was simple to find the value of k (13) which gives a quadratic with equal roots when shifted to the LHS. 3-13=10.

Answer 14

\[-10x ^{2}+20x-10=(-5x+5)(2x-2)\]
The roots are 1,1

Answer 15

http://www.wolframalpha.com/input/?i=solve+5%284x-2x%5E2%29%2B3+-+k+%3D+0

Answer 16

Sorry for the typo above. 3-13=-10

Answer 17

WolframAlpha indicates that k=13 will give roots of 1,1. Can anyone see something wrong with my method of arriving at the same result?

Answer 18

is 13 the right answer? 13 is what i got with my interpretation of the problem

Answer 19

equal roots mean the discriminant = 0 (double root)
so just set b^2-4ac = 0.