The positive difference between the two roots of the quadratic equation $3x^2 - 7x - 8 = 0$ can be written as $\frac{\sqrt{m}}{n

Question

2 months ago

8

}$, where $n$ is an integer and $m$ is an integer not divisible by the square of any prime number. Find $m + n$.

2 answers:

Svet_ta [12.7K] · Answer 1 · 2026-03-29T10:30:46+03:00

Answer: 148

Step-by-step explanation:

3x² - 7x - 8 = 0

Since we need to calculate square roots, we'll bypass the factoring step and utilize the quadratic formula,

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

145 = 5×29, confirming no square factors exist. The positive difference is

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

thus m=145, n=3, giving a total of

Svet_ta [12.7K] · Answer 2 · 2026-03-29T10:31:51+03:00

3x² - 7x - 8 = 0

Since we need to calculate square roots, we'll skip factoring and directly apply the quadratic formula,

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

145 = 5×29, indicating there are no square factors. The positive difference is

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

therefore, m=145, n=3 leading to a sum of

Answer: 148