Answer :
We start with the equation
[tex]$$
4x^3 + 48x = 28x^2.
$$[/tex]
Step 1. Move all terms to one side.
Subtract [tex]$28x^2$[/tex] from both sides to obtain
[tex]$$
4x^3 + 48x - 28x^2 = 0.
$$[/tex]
Rearrange the terms in descending order:
[tex]$$
4x^3 - 28x^2 + 48x = 0.
$$[/tex]
Step 2. Factor out the greatest common factor (GCF).
All terms have a common factor of [tex]$4x$[/tex]. Factoring [tex]$4x$[/tex] out:
[tex]$$
4x^3 - 28x^2 + 48x = 4x\left(x^2 - 7x + 12\right) = 0.
$$[/tex]
Step 3. Factor the quadratic expression.
Factor the quadratic [tex]$x^2 - 7x + 12$[/tex]. Find two numbers that multiply to [tex]$12$[/tex] and add to [tex]$-7$[/tex]. These numbers are [tex]$-3$[/tex] and [tex]$-4$[/tex]. Hence,
[tex]$$
x^2 - 7x + 12 = (x - 3)(x - 4).
$$[/tex]
Step 4. Write the complete factorization.
Substitute back into the factored expression:
[tex]$$
4x\,(x - 3)(x - 4) = 0.
$$[/tex]
Step 5. Solve for [tex]$x$[/tex].
Set each factor equal to zero:
1. From [tex]$4x=0$[/tex], we get:
[tex]$$
x = 0.
$$[/tex]
2. From [tex]$x - 3=0$[/tex], we get:
[tex]$$
x = 3.
$$[/tex]
3. From [tex]$x - 4=0$[/tex], we get:
[tex]$$
x = 4.
$$[/tex]
Final Answer:
The solutions to the equation are
[tex]$$
x = 0,\quad x = 3,\quad \text{and} \quad x = 4.
$$[/tex]
[tex]$$
4x^3 + 48x = 28x^2.
$$[/tex]
Step 1. Move all terms to one side.
Subtract [tex]$28x^2$[/tex] from both sides to obtain
[tex]$$
4x^3 + 48x - 28x^2 = 0.
$$[/tex]
Rearrange the terms in descending order:
[tex]$$
4x^3 - 28x^2 + 48x = 0.
$$[/tex]
Step 2. Factor out the greatest common factor (GCF).
All terms have a common factor of [tex]$4x$[/tex]. Factoring [tex]$4x$[/tex] out:
[tex]$$
4x^3 - 28x^2 + 48x = 4x\left(x^2 - 7x + 12\right) = 0.
$$[/tex]
Step 3. Factor the quadratic expression.
Factor the quadratic [tex]$x^2 - 7x + 12$[/tex]. Find two numbers that multiply to [tex]$12$[/tex] and add to [tex]$-7$[/tex]. These numbers are [tex]$-3$[/tex] and [tex]$-4$[/tex]. Hence,
[tex]$$
x^2 - 7x + 12 = (x - 3)(x - 4).
$$[/tex]
Step 4. Write the complete factorization.
Substitute back into the factored expression:
[tex]$$
4x\,(x - 3)(x - 4) = 0.
$$[/tex]
Step 5. Solve for [tex]$x$[/tex].
Set each factor equal to zero:
1. From [tex]$4x=0$[/tex], we get:
[tex]$$
x = 0.
$$[/tex]
2. From [tex]$x - 3=0$[/tex], we get:
[tex]$$
x = 3.
$$[/tex]
3. From [tex]$x - 4=0$[/tex], we get:
[tex]$$
x = 4.
$$[/tex]
Final Answer:
The solutions to the equation are
[tex]$$
x = 0,\quad x = 3,\quad \text{and} \quad x = 4.
$$[/tex]
