How do you know how many solutions #2x^2+5x-7=0# has?

2 Answers

Answer:

The roots are #x=-7/2# and #x=1#
graph{2x^2+5x-7 [-20, 20, -12,12] [-20, 20, -12, 12]}

Explanation:

One way to find the number of roots is by the graph. It is clear that the graph crosses the x-axis at 2 different values of x. Therefore there are 2 roots.

graph{2x^2+5x-7 [-20, 20, -12,12] [-20, 20, -12, 12]}

The give equation is
#2x^2+5x-7=0#
By factoring method,
#2x^2+5x-7=0#
#(2x+7)(x-1)=0#
by the zero property
#2x+7=0# and #x-1=0#
it follows
the roots are
#x=-7/2# and #x=1#

It can also be checked from the graph the points #(-7/2, 0)# and #(1, 0)#
God bless...I hope the explanation is useful.

Mar 2, 2018

Answer:

Using the quadratic formula, you can find out that the quadratic has two real solutions.

Explanation:

By evaluating the discriminant from the quadratic formula (#b^2-4ac#), we can find out if the quadratic has two, one, or no real solutions.

If the discriminant is greater than #0#, that means that the quadratic has #2# real solutions.

Furthermore, if the discriminant is greater than #0# and is a perfect square, the quadratic has two real and rational solutions.

If the discriminant is exactly #0#, then the quadratic has exactly #1# real solution.

Lastly, if the discriminant is less than #0#, then the quadratic does not have any real solutions.

Let's evaluate the discriminant for our quadratic:

#color(white)=>b^2-4ac#

#=>5^2-(4(2)(-7))#

#=25-(8(-7))#

#=25-(-56)#

#=25+56#

#=81#

Since the discriminant is greater than #0#, the quadratic has two real solutions. Also, since it's a perfect square, then two solutions are also rational.