Question

How do you find the nature of the roots using the discriminant given `5 - 7x^2 + 2x = 0`?

Answer 1

Because the discriminant is 144>0, there are two real roots.

Explanation:

First, rewrite the formula in standard form, `ax^2+bx+c=0`
`-7x^2+2x+5=0`

Note that
`a=-7`
`b=2`
`c=5`

The discriminant is given as `b^2-4ac` from the quadratic formula. The nature of the roots can be determined by knowing the three rules for the discriminant.

• If `b^2-4ac=0`, then there is one solution.
• If `b^2-4ac>0`, then there are two solutions.
• If If `b^2-4ac<0`, then there are no real solutions.

Plugging `a`, `b`, and `c` from above into the discriminant gives
`(2)^2-4(-7)(5)=4+28(5)=144`

Because the discriminant is 144>0, there are two solutions. The parabola has roots on the `x`-axis at two points. Finally, because the coefficient `a=-7` is negative, the parabola opens downward, as in the following graph:

graph{y=-7x^2+2x+5 [-1.3, 1.5, -2, 5.5]}