Question

How do you find the value of the discriminant and state the type of solutions given `9n^2-3n-8=-10`?

Answer 1

`Delta=-63`
roots are not real.

Explanation:

The `color(blue)"discriminant"`

`color(red)(|bar(ul(color(white)(a/a)color(black)(Delta=b^2-4ac)color(white)(a/a)|)))`
where a, b and c are the coefficients in the standard quadratic equation. The value of the discriminant provides information on the nature of the roots.

`color(orange)"Reminder" color(red)(|bar(ul(color(white)(a/a)color(black)(ax^2+bx+c=0)color(white)(a/a)|)))`

`•b^2-4ac>0tocolor(blue)"roots are real and irrational"`

`•b^2-4ac>0" and a square"tocolor(blue)"roots are real and rational"`

`•b^2-4ac=0tocolor(blue)"roots are real/rational and equal"`

`•b^2-4ac<0tocolor(blue)"roots are not real"`

Equate the given equation to zero.

`rArr9n^2-3n+2=0`

here a = 9 ,b =- 3 and c=2

`rArrb^2-4ac=(-3)^2-(4xx9xx2)=-63<0`

Since discriminant is less than zero then roots of the quadratic equation are not real.

Solving the equation using the `color(red)"quadratic formula"`

`x=(3±sqrt(-63))/18=3/18±(3isqrt7)/18=1/6+-1/6isqrt7`

The roots of the equation are not real. They are complex.