How do you find the discriminant and how many and what type of solutions does x^2 + 10x + 25 = 64 have?

1 Answer
May 8, 2018

The quadratic has two real solutions. (The solutions are x=-13 and x=3.)

Explanation:

To find the discriminant, first bring everything to one side of the equation:

x^2+10x+25=64

x^2+10x-39=0

Now, use the a, b, and c values (1, 10, and -39) and plug them into the discriminant of the quadratic formula (highlighted in red):

x=(-b+-sqrtcolor(red)(b^2-4ac))/(2a)

Discriminant:

color(white)=b^2-4ac

=(10)^2-4(1)(-39)

=100+156

=256

Since the discriminant is positive, the quadratic has two real solutions.

To solve for the solutions, you can use the quadratic formula:

x=(-10+-sqrt(10^2-4(1)(39)))/(2(1))

x=(-10+-16)/2

x=-5+-8

x=-13,3

Or you could factor the quadratic:

x^2+10x-39=0

(x+13)(x-3)=0

x=-13,3

Or you could take the original problem and square root both sides:

x^2+10x+25=64

(x+5)(x+5)=64

(x+5)^2=64

x+5=+-sqrt64

x+5=+-8

x=+-8-5

x=-13,3

Hope this helped!