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

1 Answer
May 11, 2017

Answer:

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]}