The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $/frac{/sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$.

Question

Mathematics

Heidy Keith

19 May, 05:50

The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $/frac{/sqrt{m}}{n}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$.

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Jensen Pham · Answer 1

3x² - 7x - 8 = 0

We're asked about square roots so we won't try to factor; we'll go right for the quadratic formula,

x = (7 ± √ (7² - 4 (3) (-8))) / (2 (3)) = (7 ± √ (49+96)) / 6 = 7/6 ± √145/6

145 = 5*29, so no square factors. The positive difference is

d = (7/6 + √145/6) - (7/6 - √145/6) = 2√145/6 = √145/3

so m=145, n=3 for a sum of

Answer: 148