Question

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 $m + n$.

Answer 1

Answer: 148

Step-by-step explanation:

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 2

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