Question

How do you find the roots, real and imaginary, of `y=-x^2+5x-9` using the quadratic formula?

Answer 1

real roots: none
imaginary roots: `x=(5+-sqrt(11)i)/2`

Explanation:

Since the given equation is already in standard form, identify the `color(blue)a,color(darkorange)b,` and `color(violet)c` values. Then plug the values into the quadratic formula to solve for the roots.

`y=color(blue)(-1)x^2` `color(darkorange)(+5)x` `color(violet)(-9)`

`color(blue)(a=-1)color(white)(XXXXX)color(darkorange)(b=5)color(white)(XXXXX)color(violet)(c=-9)`

`x=(-b+-sqrt(b^2-4ac))/(2a)`

`x=(-(color(darkorange)(5))+-sqrt((color(darkorange)(5))^2-4(color(blue)(-1))(color(violet)(-9))))/(2(color(blue)(-1)))`

`x=(-5+-sqrt(25-36))/-2`

`x=(-5+-sqrt(-11))/-2`

`x=(-(-5+sqrt(11)i))/2color(white)(X),color(white)(X)(-(-5-sqrt(11)i))/2`

`x=(5-sqrt(11)i)/2color(white)(X),color(white)(X)(5+sqrt(11)i)/2`

`color(green)(|bar(ul(color(white)(a/a)x=(5+-sqrt(11)i)/2color(white)(a/a)|)))`

`:.`, the imaginary roots are `x=(5+-sqrt(11)i)/2`.