Question

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

6x2 + 13x + 6 = 0

Answer 1

A positive discriminant shows that the quadratic equation
`6x^2+13x+6=0` has two distinct real number solutions namely
`x_1=(-2)/(3)` or
`x_2=(3)/(2)`

Explanation:

Consider the given Quadratic equation
`6x^2+13x+6=0`

Discriminant is a part of quadratic formula that shows the nature of root for a given quadratic equation.

Quadratic formula for a general quadratic equation of the form
`ax^2+bx+c=0` is given as:

`x=(-b\pm√(b^2-4ac))/(2a)` ......(A)

Where,
`√(b^2-4ac)` is the discriminant. .......(1)

The discriminant can be positive, zero, or negative.

A positive discriminant shows that the quadratic equation has two distinct real number solutions.

A discriminant of zero shows that the quadratic equation has a repeated real number solution.

A negative discriminant shows that neither of the solutions are real numbers.

Given Quadratic equation
`6x^2+13x+6=0`

Here, a= 6 , b=13, c= 6

Put in (1) ,
`√(b^2-4ac)`

`\Rightarrow √((13)^2-4 * 6 * 6)`

`\Rightarrow √(169-144)`

`\Rightarrow √(25)`

`\Rightarrow 5`

Thus, a positive discriminant shows that the quadratic equation has two distinct real number solutions.

Roots can be find as, Using (A)

`x=(-b\pm√(b^2-4ac))/(2a)`

`x=(-13\pm 5)/(2 * 6)`

`x_1=(-13+ 5)/(12)` or
`x_2=(-13- 5)/(12)`

`x_1=(-8)/(12)` or
`x_2=(18)/(12)`

`x_1=(-2)/(3)` or
`x_2=(3)/(2)`

Hence, the system has two distinct real roots.