# How do you use the discriminant to determine the nature of the solutions given  2x^2+4x+1=0?

Jul 20, 2016

roots are real and irrational.

#### Explanation:

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder}}$

Rational numbers $\mathbb{Q}$ are numbers which can be written in the form $\frac{a}{b}$ where a and b are integers $\mathbb{Z}$

Numbers such as $\frac{5}{2} , - \frac{2}{3} , 6 = \frac{6}{1} \text{ are rational}$ otherwise they are irrational, $4. \overline{6} , \sqrt{5} , \pi \text{ etc}$ are irrational.

The discriminant $\Delta = {b}^{2} - 4 a c$ informs us about the nature of the roots.

•b^2-4ac>0rArrcolor(blue)"roots are real and irrational"

•b^2-4ac>0" and a square"rArrcolor(blue)" roots are real and rational"

•b^2-4ac=0rArrcolor(blue)" roots are real/rational and equal"

•b^2-4ac<0rArrcolor(blue)"roots are not real"

For $2 {x}^{2} + 4 x + 1 = 0 \Rightarrow a = 2 , b = 4 , c = 1$

$\Rightarrow {b}^{2} - 4 a c = {4}^{2} - \left(4 \times 2 \times 1\right) = 16 - 8 = 8 > 0$

Since discriminant > 0 , roots are real and irrational.
$\textcolor{red}{\text{-----------------------------------------------------------}}$

As a check for you,let's solve the equation using the $\textcolor{m a \ge n t a}{\text{ quadratic formula}}$

color(red)(|bar(ul(color(white)(a/a)color(black)(x=(-b±sqrt(b^2-4ac))/(2a))color(white)(a/a)|)))
Using the values of a , b and c from above.

rArrx=(-4±sqrt8)/4

The roots are x = -0.293 and x = -1.707 (to 3 decimal places)

Thus roots are real and irrational as predicted.
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