Question

Use the discriminant to describe the roots of each equation. Then select the best description.

2 = x2 + 5x

double root
real and rational root
real and irrational root
imaginary root

Answer 1

x2+5x-2=0
a=1, b=5, c=-2

b2-4ac is the discriminant.
(5^2)-4(1)(-2)
25+8=33, so the discriminant is positive

This means there are two real rational roots, so the answer is the second option

Answer 2

Step-by-step explanation

• Given the equation.

`2 = {x}^(2) + 5x`

• Arrange the expression into the standard form.

`0 = {x}^(2) + 5x - 2 \\ {x}^(2) + 5x - 2 = 0`

• Discriminant

From the standard form equation and from the given equation.

`a {x}^(2) + bx + c = 0 \\ a = 1 \: \: \: b = 5 \: \: \: c = - 2`

Discriminant Formula

`D = {b}^(2) - 4ac`

Substitute the value of terms in the discriminant formula.

`D = {(5)}^(2) - 4(1)( - 2) \\ D = 25 - 4( - 2) \\ D = 25 + 8 \\ D = 33`

Discriminant Types

`D > 0 \: \: \: \: 2 \: \: real \: \: roots \\ D = 0 \: \: \: \: 1 \: \: real \: \: root \\ D < 0 \: \: \: imaginary \: \: roots`

We can rule out the Imaginary root and double root because the discriminant is greater than 0. Therefore, the only choices are either Irrational or Rational.

However, we cannot factor the expression by using two brackets or common factor. We can say that the equation has two real irrational roots.

Answer

• real and irrational root