78th Problem 2016

Algebra Level 2

What is the nature of the roots of this equation:

2 x 2 − 4 x + 7 = 2 { 2x }^{ 2 }-4x+7=2

Check out the set: 2016 Problems

Real and Distinct Not Real Real and Equal

Angela Fajardo by Angela Fajardo , 19, Philippines

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2 solutions

Nikhil Raj
Jun 5, 2017

2 x 2 − 4 x + 7 = 2 2 x 2 − 4 x + 5 = 0 The discriminant will determine, which kind of roots does given equation has. D = b 2 − 4 a c = 16 − 40 = − 24 < 0 Since, the discriminant is negative, so the roots will be imaginary, hence not real \begin{array}{c}2x^2 - 4x + 7 = 2 \\ 2x^2 - 4x + 5 = 0 \\ {\text{The discriminant will determine, which kind of roots does given equation has.}} \\ D = b^2 - 4ac = 16 - 40 = -24 < 0 \\ {\text{Since, the discriminant is negative, so the roots will be imaginary, hence}} \color{#EC7300}{\boxed{{\text{not real}}}} \end{array}

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Emanuel Dicker
May 4, 2016

In a standard a^2x + bx + c equation, the discriminant of the equation determines the type of roots that equation has.

The discriminant is given by b^2 -4ac

In our equation 2x^2 -4x +5,

a = 2, b= -4, c= 5

b^2 - 4ac= (-4)^2 -(4 x 5 x 2) =

16 - 40= -24

Thus, since the number is below zero, the roots are imaginary.

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Moderator note:

Simple standard approach using the discriminant.

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