Certainly! Let's solve the given quadratic equation [tex]\( x^2 + 5x = 0 \)[/tex] step by step.
First, we start with the given equation:
[tex]\[ x^2 + 5x = 0 \][/tex]
This is a quadratic equation, and there are several methods to solve it. In this case, we'll use the factoring method. Here's the detailed solution:
1. Factor out the common term:
Notice that both terms on the left side of the equation have a common factor of [tex]\( x \)[/tex]. We can factor [tex]\( x \)[/tex] out:
[tex]\[
x(x + 5) = 0
\][/tex]
2. Set each factor equal to zero:
According to the zero-product property, if a product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
x = 0
\][/tex]
and
[tex]\[
x + 5 = 0
\][/tex]
3. Solve the simple equations:
- The first equation [tex]\( x = 0 \)[/tex] is already solved.
- For the second equation [tex]\( x + 5 = 0 \)[/tex], we solve for [tex]\( x \)[/tex] by isolating the variable:
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
So, the solutions to the equation [tex]\( x^2 + 5x = 0 \)[/tex] are:
[tex]\[
x = 0 \quad \text{and} \quad x = -5
\][/tex]
Thus, the solutions are [tex]\( \boxed{0 \text{ and } -5} \)[/tex].
