Sure, let's solve the equation [tex]\(x^3 - 7x^2 = -12x\)[/tex] step by step.
1. Rewrite the equation: Start by moving all terms to one side of the equation so that it is set equal to zero.
[tex]\[ x^3 - 7x^2 + 12x = 0 \][/tex]
2. Factor out the greatest common factor (GCF): Notice that each term in the equation has a common factor of [tex]\(x\)[/tex]. Factor [tex]\(x\)[/tex] out of the equation:
[tex]\[ x(x^2 - 7x + 12) = 0 \][/tex]
3. Solve for the factors: Now, solve for [tex]\(x\)[/tex] by setting each factor equal to zero.
- The first factor is [tex]\(x\)[/tex]:
[tex]\[ x = 0 \][/tex]
- For the quadratic factor [tex]\(x^2 - 7x + 12 = 0\)[/tex], we will solve using factoring. We look for two numbers that multiply to 12 (the constant term) and add to -7 (the coefficient of the linear term).
The pair of numbers that work are -3 and -4 because:
[tex]\[ (-3) \cdot (-4) = 12 \][/tex]
[tex]\[ (-3) + (-4) = -7 \][/tex]
4. Factor the quadratic expression:
[tex]\[ x^2 - 7x + 12 = (x - 3)(x - 4) \][/tex]
So our equation becomes:
[tex]\[ x(x - 3)(x - 4) = 0 \][/tex]
5. Solve for each factor: Set each factor equal to zero and solve for [tex]\(x\)[/tex].
- For [tex]\(x = 0\)[/tex]:
[tex]\[ x = 0 \][/tex]
- For [tex]\(x - 3 = 0\)[/tex]:
[tex]\[ x = 3 \][/tex]
- For [tex]\(x - 4 = 0\)[/tex]:
[tex]\[ x = 4 \][/tex]
7. List all solutions: The solutions to the equation [tex]\(x^3 - 7x^2 = -12x\)[/tex] are:
[tex]\[ x = 0, 3, 4 \][/tex]
So the solutions to the equation are [tex]\(\boxed{0, 3, 4}\)[/tex].
