We need to determine which of the given values of [tex]\( x \)[/tex] are the roots of the equation [tex]\( 4x(x - 3) = 2x + 6 \)[/tex]. The values of [tex]\( x \)[/tex] to test are 3, 0, and 2.
The first step is to rewrite the given equation in standard quadratic form.
### Step 1: Rewrite the equation
Starting with the provided equation:
[tex]\[ 4x(x - 3) = 2x + 6 \][/tex]
Distribute the [tex]\( 4x \)[/tex] on the left-hand side:
[tex]\[ 4x^2 - 12x = 2x + 6 \][/tex]
Next, bring all terms to one side of the equation to set it to zero:
[tex]\[ 4x^2 - 12x - 2x - 6 = 0 \][/tex]
Combine like terms:
[tex]\[ 4x^2 - 14x - 6 = 0 \][/tex]
We now have the quadratic equation:
[tex]\[ 4x^2 - 14x - 6 = 0 \][/tex]
### Step 2: Test the given values
We test each value of [tex]\( x \)[/tex] by substituting them into the quadratic equation and checking if they satisfy the equation.
#### Test [tex]\( x = 3 \)[/tex]:
[tex]\[ 4(3)^2 - 14(3) - 6 = 0 \][/tex]
[tex]\[ 4(9) - 42 - 6 = 0 \][/tex]
[tex]\[ 36 - 42 - 6 = 0 \][/tex]
[tex]\[ -12 \neq 0 \][/tex]
So, [tex]\( x = 3 \)[/tex] is not a root.
#### Test [tex]\( x = 0 \)[/tex]:
[tex]\[ 4(0)^2 - 14(0) - 6 = 0 \][/tex]
[tex]\[ 4(0) - 14(0) - 6 = 0 \][/tex]
[tex]\[ 0 - 6 = 0 \][/tex]
[tex]\[ -6 \neq 0 \][/tex]
So, [tex]\( x = 0 \)[/tex] is not a root.
#### Test [tex]\( x = 2 \)[/tex]:
[tex]\[ 4(2)^2 - 14(2) - 6 = 0 \][/tex]
[tex]\[ 4(4) - 14(2) - 6 = 0 \][/tex]
[tex]\[ 16 - 28 - 6 = 0 \][/tex]
[tex]\[ -18 \neq 0 \][/tex]
So, [tex]\( x = 2 \)[/tex] is not a root.
### Conclusion
None of the given values [tex]\( x = 3 \)[/tex], [tex]\( x = 0 \)[/tex], or [tex]\( x = 2 \)[/tex] are roots of the equation [tex]\( 4x(x - 3) = 2x + 6 \)[/tex].
Thus, the determined roots from the given values are:
[tex]\[
\boxed{[]}
\][/tex]
There are no values among 3, 0, and 2 that satisfy the equation.
