Question

What is the value of k in the equation, so that one root exceeds the other by 4

Answer 1

Answer:   k = -15

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Explanation:

Let p(x) = 4x^2+4x+k be the polynomial function.

Also, let r and s be the two roots of the polynomial p(x).

By definition of what it means to be a root, we know that

p(r) = 0

p(s) = 0

So this means p(r) = p(s).

Because one root exceeds another by 4, we can say s = r+4.

So the equation p(r) = p(s) updates to p(r) = p(r+4).

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Let's compute p(r) and p(r+4)

So,

p(x) = 4x^2+4x+k

p(r) = 4r^2+4r+k

and

p(x) = 4x^2+4x+k

p(r+4) = 4(r+4)^2+4(r+4)+k

p(r+4) = 4(r^2+8r+16)+4(r+4)+k

p(r+4) = 4r^2+32r+64+4r+16+k

p(r+4) = 4r^2+36r+80+k

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Now equate those results

p(r) = p(r+4)

4r^2+4r+k = 4r^2+36r+80+k

4r+k = 36r+80+k        ...... the 4r^2 terms cancel

4r = 36r+80                ..... the k terms cancel as well

4r-36r = 80

-32r = 80

r = 80/(-32)

r = (16*5)/(-16*2)

r = -5/2 = -2.5 is one of the roots

s = r+4

s = -2.5+4

s = 1.5 = 3/2 is the other root.

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With this in mind, we can use either r or s to find the value of k

p(x) = 4x^2 + 4x + k

p(r) = 4r^2 + 4r + k

p(r) = 4(-2.5)^2 + 4(-2.5) + k

p(r) = 15+k

0 = 15+k

k+15 = 0

k = -15

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To confirm this answer, you can use the quadratic formula to solve 4x^2+4x-15 = 0. You should get the two roots r = -5/2 = -2.5 and s = 3/2 = 1.5

Then note how s-r = 4 which is the same as saying s = r+4.