Q. \(x^2 – 9x = 0\)

Q: What are the solutions to x(x-9)=0?

Set each factor to zero: x=0 or x-9=0, so x=9.

Q: How can you solve it by completing the square?

Rewrite: x^2-9x= x^2-9x+\left(\frac{9}{2}\right)^2-\left(\frac{9}{2}\right)^2. Then solve \left(x-\frac{9}{2}\right)^2=\left(\frac{9}{2}\right)^2, giving x=0,9.

Q: What is the factoring approach using the greatest common factor (GCF)?

The GCF is x. Divide: x^2-9x = x(x-9). Solve x=0 or x-9=0.

Q: How do the roots relate to a quadratic’s axis and symmetry?

For x^2-9x, axis is x=\frac{9}{2}. The roots 0 and 9 are symmetric around 4.5.

Q: How do you verify the answers quickly?

Substitute x=0: 0^2-9\cdot0=0. Substitute x=9: 81-81=0. Both satisfy the equation.

Answer

Solve \(x^2-9x=0\) by factoring:

\[
x^2-9x = x(x-9)=0
\]

So \(x=0\) or \(x-9=0\), giving \(x=9\).

Final results: \(x=0\) or \(x=9\).

Detailed Explanation

We want to solve the equation \(x^2 – 9x = 0\).

Step 1: Set up the equation in factored form.

Notice that both terms contain a factor of \(x\). Factor out \(x\) from the left-hand side:

\[
x^2 – 9x = x(x – 9)
\]

So the equation becomes:

\[
x(x – 9) = 0
\]

Step 2: Use the Zero Product Property.

The Zero Product Property says: if \(a b = 0\), then \(a = 0\) or \(b = 0\).

Here, \(a = x\) and \(b = x – 9\). Therefore:

\[
x = 0 \quad \text{or} \quad x – 9 = 0
\]

Step 3: Solve each equation.

Case 1: \(x = 0\)

Case 2: \(x – 9 = 0\)

Add \(9\) to both sides:

\[
x = 9
\]

Final Answer:

The solutions to \(x^2 – 9x = 0\) are:

\[
x = 0,\; 9
\]

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Algebra FAQ

How do you factor \(x^2-9x=0\)?

Factor out \(x\): \(x^2-9x=x(x-9)=0\).

What are the solutions to \(x(x-9)=0\)?

Set each factor to zero: \(x=0\) or \(x-9=0\), so \(x=9\).

How can you solve it by completing the square?

Rewrite: \(x^2-9x= x^2-9x+\left(\frac{9}{2}\right)^2-\left(\frac{9}{2}\right)^2\). Then solve \(\left(x-\frac{9}{2}\right)^2=\left(\frac{9}{2}\right)^2\), giving \(x=0,9\).

What is the factoring approach using the greatest common factor (GCF)?

The GCF is \(x\). Divide: \(x^2-9x = x(x-9)\). Solve \(x=0\) or \(x-9=0\).

How do the roots relate to a quadratic’s axis and symmetry?

For \(x^2-9x\), axis is \(x=\frac{9}{2}\). The roots \(0\) and \(9\) are symmetric around \(4.5\).

How do you verify the answers quickly?

Substitute \(x=0\): \(0^2-9\cdot0=0\). Substitute \(x=9\): \(81-81=0\). Both satisfy the equation.

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