How do you find the discriminant and how many and what type of solutions does x^2 + 10x + 25 = 64x2+10x+25=64 have?

1 Answer
Aviv S.
May 8, 2018

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

Explanation:

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

x^2+10x+25=64x2+10x+25=64

x^2+10x-39=0x2+10x−39=0

Now, use the aa, bb, and cc values (11, 1010, and -39−39) and plug them into the discriminant of the quadratic formula (highlighted in red):

x=(-b+-sqrtcolor(red)(b^2-4ac))/(2a)x=−b±√b2−4ac2a

Discriminant:

color(white)=b^2-4ac=b2−4ac

=(10)^2-4(1)(-39)=(10)2−4(1)(−39)

=100+156=100+156

=256=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±√102−4(1)(39)2(1)

x=(-10+-16)/2x=−10±162

x=-5+-8x=−5±8

x=-13,3x=−13,3

Or you could factor the quadratic:

x^2+10x-39=0x2+10x−39=0

(x+13)(x-3)=0(x+13)(x−3)=0

x=-13,3x=−13,3

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

x^2+10x+25=64x2+10x+25=64

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

(x+5)^2=64(x+5)2=64

x+5=+-sqrt64x+5=±√64

x+5=+-8x+5=±8

x=+-8-5x=±8−5

x=-13,3x=−13,3

Hope this helped!

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