How do you find the value of the discriminant and determine the nature of the roots #4a^2=8a-4#?

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Answer 1 · 2018-04-02T10:16:21

See a solution process below:

First, put this equation in standard quadratic form:

#4a^2 = 8a - 4#

#4a^2 - color(red)(8a) + color(blue)(4) = 8a - color(red)(8a) - 4 + color(blue)(4)#

#4a^2 - 8a + 4 = 0 - 0#

#4a^2 - 8a + 4 = 0#

For #ax^2 + bx + c = 0#, the values of #x# which are the solutions to the equation are given by:

#x = (-b +- sqrt(b^2 - 4ac))/(2a)#

The discriminate is the portion of the quadratic equation within the radical: #color(blue)(b)^2 - 4color(red)(a)color(green)(c)#

If the discriminate is:
- Positive, you will get two real solutions
- Zero you get just ONE solution
- Negative you get complex solutions

To find the discriminant for this problem substitute:

#color(red)(4)# for #color(red)(a)#

#color(blue)(-8)# for #color(blue)(b)#

#color(green)(4)# for #color(green)(c)#

#color(blue)(-8)^2 - (4 * color(red)(4) * color(green)(4)) =>#

#64 - 64 =>#

#0#

Because the discriminate is #0# there will be just one solution or one root.