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

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Answer 1 · 2018-05-08T23:23:31

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

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!