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I need some help, Im trying to solve the below but to no avail, would appreciate your guidance :)

Find the value of $k$ if the roots of $$3x^2+5x-k=0,$$ differ by two.

Chinny84
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user2509708
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  • Write down the two roots in terms of $k$ then equate the absolute value of the difference to $2$. – dxiv Apr 21 '17 at 18:56

4 Answers4

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Outline:

Let the roots be $u,v$.

Use Vieta's formulas to find $u+v$ and $uv$.

Then use the identity $(u-v)^2 = (u+v)^2 - 4uv$.

Once you have the value of $(u-v)^2$ in terms of $k$, set it to $4$ and solve for $k$.

quasi
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Note that the polynomial factors as $(3x+a)(x+b)$. Then the roots are $a/3$ and $b$, which have to differ by two. If you multiply $(3x+a)(x+b)$ out and compare it to the original polynomial, you should have enough relationships to extract the answer.

B. Goddard
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Let $r $ and $r+2$ be the roots.

Their sum is $$r+ (r+2)=\frac{-5}{3}$$

thus

$$r=\frac {-11}{6}. $$

Their product is

$$r (r+2)=\frac {-k}{3}.$$

thus $$k=\frac{11}{12} $$

the roots are $-\frac {11}{6} $ and $\frac 1 6$

hamam_Abdallah
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From the quadratic formula, we know that if the equation $ax^2+bx+c=0$ has two distinct real roots, then their difference is $2\sqrt{b^2-4ac}$. Plugging in the values from your equation, this means that $2\sqrt{25+12k}=2$. Solve for $k$.

amd
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