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

May 11, 2017

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

Explanation:

First, rewrite the formula in standard form, $a {x}^{2} + b x + c = 0$
$- 7 {x}^{2} + 2 x + 5 = 0$

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

The discriminant is given as ${b}^{2} - 4 a c$ from the quadratic formula. The nature of the roots can be determined by knowing the three rules for the discriminant.

• If ${b}^{2} - 4 a c = 0$, then there is one solution.
• If ${b}^{2} - 4 a c > 0$, then there are two solutions.
• If If ${b}^{2} - 4 a c < 0$, then there are no real solutions.

Plugging $a$, $b$, and $c$ from above into the discriminant gives
${\left(2\right)}^{2} - 4 \left(- 7\right) \left(5\right) = 4 + 28 \left(5\right) = 144$

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]}