Question

If x = −3 and x = `2/3` are solutions of quadratic equation mx² + 7x + n = 0, find the values of m and n.

Answer 1

Step 1: Sum and product of roots

Sum of roots = `-b/a, "Product of roots" = c/a` 

• Roots: x₁ = −3, x₂ = `2/3`
• Coefficients: a = m, b = 7, c = n

Step 2: Calculate sum and product of roots

`x_1 + x_2 = -3 + 2/3 = -9/3 + 2/3 = -7/3`

Using Sum of roots = `−b/a`

`-7/3 = -7/m`

Equating the numerators, m = 3

Product of roots: `x_1xxx_2 = (-3)xx 2/3 = -2`

Using Product of roots = `c/a`

`-2 = n/m`

Substitute m = 3

`-2 = n/3 => n=-6`

m = 3, n = −6