Answer :
To solve the equation [tex]\(5x^3 + 45x^2 + 70x = 0\)[/tex], we can start by factoring it. Here’s the step-by-step process:
1. Factor out the greatest common factor (GCF):
- Notice that each term in the equation has a common factor of [tex]\(5x\)[/tex]. So, we can factor [tex]\(5x\)[/tex] out of the entire equation:
[tex]\[
5x(x^2 + 9x + 14) = 0
\][/tex]
2. Apply the Zero Product Property:
- According to the Zero Product Property, if a product of factors is zero, at least one of the factors must be zero.
- This gives us two possible equations to solve:
[tex]\[
5x = 0
\][/tex]
[tex]\[
x^2 + 9x + 14 = 0
\][/tex]
3. Solve the first equation:
- [tex]\(5x = 0\)[/tex]
- Divide both sides by 5:
[tex]\[
x = 0
\][/tex]
4. Solve the quadratic equation:
- [tex]\(x^2 + 9x + 14 = 0\)[/tex]
- We can factor this quadratic equation:
[tex]\[
x^2 + 9x + 14 = (x + 7)(x + 2) = 0
\][/tex]
- Set each factor to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
5. List all solutions:
- The solutions to the equation [tex]\(5x^3 + 45x^2 + 70x = 0\)[/tex] are:
[tex]\[
x = 0, \quad x = -7, \quad x = -2
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the original equation.
1. Factor out the greatest common factor (GCF):
- Notice that each term in the equation has a common factor of [tex]\(5x\)[/tex]. So, we can factor [tex]\(5x\)[/tex] out of the entire equation:
[tex]\[
5x(x^2 + 9x + 14) = 0
\][/tex]
2. Apply the Zero Product Property:
- According to the Zero Product Property, if a product of factors is zero, at least one of the factors must be zero.
- This gives us two possible equations to solve:
[tex]\[
5x = 0
\][/tex]
[tex]\[
x^2 + 9x + 14 = 0
\][/tex]
3. Solve the first equation:
- [tex]\(5x = 0\)[/tex]
- Divide both sides by 5:
[tex]\[
x = 0
\][/tex]
4. Solve the quadratic equation:
- [tex]\(x^2 + 9x + 14 = 0\)[/tex]
- We can factor this quadratic equation:
[tex]\[
x^2 + 9x + 14 = (x + 7)(x + 2) = 0
\][/tex]
- Set each factor to zero and solve for [tex]\(x\)[/tex]:
[tex]\[
x + 7 = 0 \quad \Rightarrow \quad x = -7
\][/tex]
[tex]\[
x + 2 = 0 \quad \Rightarrow \quad x = -2
\][/tex]
5. List all solutions:
- The solutions to the equation [tex]\(5x^3 + 45x^2 + 70x = 0\)[/tex] are:
[tex]\[
x = 0, \quad x = -7, \quad x = -2
\][/tex]
These are the values of [tex]\(x\)[/tex] that satisfy the original equation.
