Question

For the quadratic equation 3x^2-18x-21=0, x1 and x2 are the roots. Find:

x1^2+x2^2
|x1-x2|

Answer 1

(x_1+x_2)=-b/a=18/3=6

x_1x_2=c/a=-21/3=-7

So

• (x_1+x_2)²-2x_1x_2=x_1^2+x2²
• x1²+x2²=(6)²-2(-7)=36+14=50

(x_1-x_2)²

• x1²+x2²-2x_1x2
• 50+14
• 64

x_1-x_2=8

Answer 2

`x{_1}^2+x{_2}^2=50`

`|x_1-x_2|=8`

Explanation:

Given equation:

`3x^2-18x-21=0`

Factor to find the roots:

`\implies 3(x^2-6x-7)=0`

`\implies x^2-6x-7=0`

`\implies x^2+x-7x-7=0`

`\implies x(x+1)-7(x+1)=0`

`\implies (x-7)(x+1)=0`

Therefore:

`\implies (x-7)=0 \implies x=7`

`\implies (x+1)=0 \implies x=-1`

Therefore, the roots of the quadratic equation are:

`x = 7,\quad x = -1`

Let
`x_1 = 7`

Let
`x_2 = -1`

`\implies x{_1}^2+x{_2}^2=7^2+(-1)^2=50`

`\implies |x_1-x_2|=|7-(-1)|=8`