Question

F(x) = -4x(2)+ 12x – 9

What is the value of the discriminant of f?
How many x-intercepts does the graph of f have?

Answer 1

For this case we have the following function:

`f (x) = - 4x ^ 2 + 12x-9`

`y = 0` we have:

`-4x ^ 2 + 12x-9 = 0`

Where:

`a = -4\\b = 12\\c = -9`

By definition, the discriminant of a quadratic equation is given by:

`d = b ^ 2-4 (a) (c)`

`d> 0:` Two different real roots

`d = 0:` Two equal real roots

`d <0`: Two different complex roots

Substituting the values we have:

`d = (12) ^ 2-4 (-4) (- 9)\\d = 144-144`

`d = 0`

We have two equal real roots.

To find the intersections with the x axis, we do
`y = 0:`

`-4x ^ 2 + 12x-9 = 0`

We apply the quadratic formula:

`x = \frac {-b \pm \sqrt {b ^ 2-4 (a) (c)}} {2a}`

Substituting the values we have:

`x = \frac {-12 \pm \sqrt {12 ^ 2-4 (-4) (- 9)}} {2 (-4)}\\x = \frac {-12 \pm \sqrt {144-144}} {- 8}\\x = \frac {-12 \pm0} {- 8}\\x = \frac {-12} {- 8}\\x = \frac {3} {2}`

The intersection with the x axis is
`(\frac {3} {2}, 0)`

Answer:

`d = 0`

The intersection with the x axis is
`(\frac {3} {2}, 0)`

Answer 2

Part A) The value of the discriminant of f is zero

Part B) The quadratic equation has only one x-intercept

Explanation:

Part A) What is the value of the discriminant of f?

we know that

In a quadratic equation of the form

`ax^(2) +bx+c=0`

the discriminant is equal to

`D=b^2-4ac`

in this problem we have

`f(x)=-4x^2+12x-9`

so

`a=-4\\b=12\\c=-9`

substitute

`D=12^2-4(-4)(-9)`

`D=0`

Part B) How many x-intercepts does the graph of f have?

we know that

In a quadratic equation

If the discriminant D is equal to zero, then the equation has only one real solution

That means

The quadratic equation has only one x-intercept