1967 AHSME Problems/Problem 35

Problem

The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is:

$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ \frac{1}{2}\qquad \textbf{(D)}\ \frac{3}{8}\qquad \textbf{(E)}\ \frac{1}{4}$

Solution 1

By Vieta's Formulas, the sum of the roots is $-\frac{-144}{64} = \frac{9}{4}$. Because the roots are in arithmetic progression, the middle root is the average of the other two roots, and is also the average of all three roots. Therefore, $\frac{\frac{9}{4}}{3} = \frac{3}{4}$ is the middle root.

The other two roots have a sum of $\frac{9}{4} - \frac{3}{4} = \frac{3}{2}$. By Vieta on the original cubic, the product of all $3$ roots is $-\frac{-15}{64} = \frac{15}{64}$, so the product of the remaining two roots is $\frac{\frac{15}{64}}{\frac{3}{4}} = \frac{5}{16}$. If the sum of the two remaining roots is $\frac{3}{2} = \frac{24}{16}$, and the product is $\frac{5}{16}$, the two remaining roots are also the two roots of $16x^2 - 24x + 5 = 0$.

The two remaining roots are thus $x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}$, and they have a difference of $\frac{2\sqrt{B^2 - 4AC}}{2A}$. Plugging in gives $\frac{\sqrt{24^2 - 4(16)(5)}}{16}$, which is equal to $1$, which is answer $\fbox{B}$.

Solution 2

Because the roots are in arithmetic progression, let the roots be $r-d, r,$ and $r+d$ with $d \geq 0$. From Vieta's Formulas, we know that $(r-d)+r+(r+d)=\tfrac{144}{64}$, so $3r=\tfrac 9 4$, and $r=\tfrac 3 4$.

By using Vieta again, we see that $(r-d)(r)(r+d)=\tfrac{15}{64}$. Substituting $r=\tfrac 3 4$ into this equation, we can now solve for $d$: \begin{align*} (\frac 3 4-d)(\frac 3 4)(\frac 3 4+d) &= \frac{15}{64} \\ \frac 3 4(\frac 9{16}-d^2) &= \frac{15}{64} \\ \frac 9{16}-d^2 &= \frac 5{16} \\ d^2 &= \frac 4{16} = \frac 1 4 \\ d &= \pm \frac 1 2 \end{align*} Because we defined $d \geq 0$, $d=\frac 1 2$. Therefore, the difference between the largest and smallest roots, $r+d$ and $r-d$, is $2d=2 \cdot \tfrac 1 2 = \boxed{\textbf{(B) }1}$.

See also

1967 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 34
Followed by
Problem 36
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