Q. \[ x^2-2x \]

Q: Find the vertex and minimum/maximum of f(x)=x^2-2x.

\[f(x)=(x-1)^2-1.\] Vertex is at (1,-1) and it has a minimum -1.

Q: Determine where x^2-2x is positive or negative.

\[x(x-2)>0 \Rightarrow x 2.\] And x(x-2)<0 for 0<x<2.

Answer

We factor the quadratic \(x^{2}-2x\) by taking out the common factor \(x\):

\[
x^{2}-2x = x(x-2)
\]

Final result: \(x(x-2)\)

Detailed Explanation

We want to simplify the expression \(x^2 – 2x\).

Step 1: Factor out the greatest common factor.

Both terms \(x^2\) and \(-2x\) share a factor of \(x\). So we factor out \(x\).

\[
x^2 – 2x = x(x – 2)
\]

Step 2: Write the final factored form.

The expression in factored form is \(x(x – 2)\).

Final Answer:

\(x^2 – 2x = x(x – 2)\)

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Algebra FAQ

Factor \(x^2-2x\).

\[x^2-2x=x(x-2).\]

Solve \(x^2-2x=0\).

\[x(x-2)=0 \Rightarrow x=0 \text{ or } x=2.\]

Simplify the expression \(x(x-2)\) back to standard form.

\[x(x-2)=x^2-2x.\]

Find the vertex and minimum/maximum of \(f(x)=x^2-2x\).

\[f(x)=(x-1)^2-1.\] Vertex is at \((1,-1)\) and it has a minimum \(-1\).

Complete the square for \(x^2-2x\).

\[x^2-2x=(x-1)^2-1.\]

Determine where \(x^2-2x\) is positive or negative.

\[x(x-2)>0 \Rightarrow x<0 \text{ or } x>2.\] And \(x(x-2)<0\) for \(0

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