How do you use the discriminant to determine the numbers of solutions of the quadratic equation #4x^2-20x + 25 = 0# and whether the solutions are real or complex?

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Answer 1 · 2018-06-18T10:37:37

one real root

root is #5/2#

Quadratic equations will have one or two roots (either real or complex roots)

Let our quadratic equation be #ax^2+bx+c=0#

And #x_1,x_2# be the roots of the equation

Then the discriminant of this quadratic equation will be #b^2-4ac#

#color(red)1.# when #b^2-4ac>0# we will have two different real roots

#x_1 and x_2#are real and unequal #(x_1!=x_2)#

#color(red)2.# when #b^2-4ac=0# we will have two equal real roots

#x_1 and x_2 # are real and #x_1=x_2#

#color(red)3.#when #b^2-4ac<0# we will have two complex roots

#x_1!=x_2#
#x_1 and x_2 # are complex roots

Comparing the given quadratic equation with the general quadratic equation

we get the values of #a,b,c# as #a=4,b=-20,c=25#

The Discriminant will be #(-20)^2-4xx4xx25=>400-400=>0#

the roots are real and equal (Only one root)

Hence the equation must be a perfect square

#4x^2-20x+25=>(2x-5)^2#

Hence the root of the equation is #2x-5=0=>x=5/2#

#x_1=x_2=5/2#