Q. \(x^2 + x = 0\)

Q: How do I factor x^2 + x = 0?

Factor out x: x^2 + x = x(x+1)=0. So either x=0 or x+1=0\Rightarrow x=-1.

Q: What are the solutions using the quadratic formula?

For x^2+x+0=0, a=1,b=1,c=0. Then x=\frac{-1\pm\sqrt{1-0}}{2}=\frac{-1\pm1}{2}, giving x=0 and x=-1.

Q: Why does one root equal zero?

Because the equation is x(x+1)=0. Any factor equal to zero solves it, and x=0 makes the whole product zero.

Q: Can I solve it by completing the square?

Write x^2+x= x^2+x+\frac{1}{4}-\frac{1}{4}=0. Then \left(x+\frac{1}{2}\right)^2=\frac{1}{4}, so x+\frac{1}{2}=\pm\frac{1}{2}, giving x=0,-1.

Q: What is the graph interpretation of x^2+x=0?

Solve x^2+x=x(x+1)=0. The intercepts are at x=0 and x=-1. These are the x-values where the parabola crosses the x-axis.

Q: How can I check my answers quickly?

Substitute x=0: 0^2+0=0. Substitute x=-1: (-1)^2+(-1)=1-1=0. Both satisfy the equation.

Answer

We solve the quadratic equation \(x^{2}+x=0\) by factoring:

\[
x^{2}+x = x(x+1)=0
\]

So \(x=0\) or \(x+1=0\), which gives \(x=-1\).

Final answer: \(x=0\) or \(x=-1\).

Detailed Explanation

We want to solve the equation

\[
x^2 + x = 0
\]

Step 1: Factor the left-hand side

The expression \(x^2 + x\) has a common factor of \(x\). Factor out \(x\):

\[
x^2 + x = x(x + 1)
\]

So the equation becomes

\[
x(x + 1) = 0
\]

Step 2: Use the zero product property

The zero product property says: if

\[
a \cdot b = 0
\]

then either \(a = 0\) or \(b = 0\).

Here, \(a = x\) and \(b = x + 1\). So we set each factor equal to zero:

\[
x = 0
\]

and

\[
x + 1 = 0
\]

Step 3: Solve each equation

First, from \(x = 0\), we get one solution:

\[
x = 0
\]

Second, solve \(x + 1 = 0\):

\[
x + 1 = 0
\]

Subtract \(1\) from both sides:

\[
x = -1
\]

Final Answer

The solutions to \(x^2 + x = 0\) are:

\[
x = 0 \quad \text{or} \quad x = -1
\]

See full solution

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Algebra FAQ

How do I factor \(x^2 + x = 0\)?

Factor out \(x\): \(x^2 + x = x(x+1)=0\). So either \(x=0\) or \(x+1=0\Rightarrow x=-1\).

What are the solutions using the quadratic formula?

For \(x^2+x+0=0\), \(a=1,b=1,c=0\). Then \(x=\frac{-1\pm\sqrt{1-0}}{2}=\frac{-1\pm1}{2}\), giving \(x=0\) and \(x=-1\).

Why does one root equal zero?

Because the equation is \(x(x+1)=0\). Any factor equal to zero solves it, and \(x=0\) makes the whole product zero.

Can I solve it by completing the square?

Write \(x^2+x= x^2+x+\frac{1}{4}-\frac{1}{4}=0\). Then \(\left(x+\frac{1}{2}\right)^2=\frac{1}{4}\), so \(x+\frac{1}{2}=\pm\frac{1}{2}\), giving \(x=0,-1\).

What is the graph interpretation of \(x^2+x=0\)?

Solve \(x^2+x=x(x+1)=0\). The intercepts are at \(x=0\) and \(x=-1\). These are the \(x\)-values where the parabola crosses the \(x\)-axis.

How can I check my answers quickly?

Substitute \(x=0\): \(0^2+0=0\). Substitute \(x=-1\): \((-1)^2+(-1)=1-1=0\). Both satisfy the equation.

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