Q. \(x^{2}-7x=0\)

Q: What are the solutions of x^2-7x=0?

Factor first: \[x^2-7x=x(x-7)=0.\] So x=0 or x=7.

Q: How do you factor x^2-7x?

Take out the common factor x: \[x^2-7x=x(x-7).\] Then set each factor equal to 0.

Q: Solve using the zero-product property.

From x(x-7)=0, the zero-product property gives x=0 or x-7=0, so x=7.

Q: What is the quadratic formula solution?

For x^2-7x+0=0: \[x=\frac{7\pm\sqrt{49}}{2}=\frac{7\pm 7}{2}.\] So x=0 or x=7.

Q: How can you interpret the solutions graphically?

Set y=x^2-7x=x(x-7). Zeros occur where the graph crosses the x-axis: at x=0 and x=7.

Q: What is the vertex and does it affect the roots?

Vertex at x=\frac{-b}{2a}=\frac{7}{2}. Roots remain solutions to x(x-7)=0, so x=0,7 regardless of the vertex position.

Answer

We solve \(x^2-7x=0\) by factoring:

\[
x^2-7x = x(x-7)=0
\]

So \(x=0\) or \(x-7=0\), which gives \(x=7\).

Final answers: \(x=0,\,7\).

Detailed Explanation

We want to solve the equation:

\[
x^2 – 7x = 0
\]

Step 1: Factor the equation.

Both terms involve \(x\), so we factor out \(x\) from the left-hand side.

\[
x^2 – 7x = x(x – 7)
\]

So the equation becomes:

\[
x(x – 7) = 0
\]

Step 2: Use the Zero Product Property.

The Zero Product Property says that if

\[
a b = 0
\]

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

Here:

\[
x(x – 7) = 0
\]

So either:

\[
x = 0
\]

or:

\[
x – 7 = 0
\]

Step 3: Solve each equation.

1) If \(x = 0\), that is one solution.

2) If \(x – 7 = 0\), add \(7\) to both sides:

\[
x = 7
\]

Final Answer:

The solutions are:

\[
x = 0 \quad \text{or} \quad x = 7
\]

See full solution

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Algebra FAQ

What are the solutions of \(x^2-7x=0\)?

Factor first: \[x^2-7x=x(x-7)=0.\] So \(x=0\) or \(x=7\).

How do you factor \(x^2-7x\)?

Take out the common factor \(x\): \[x^2-7x=x(x-7).\] Then set each factor equal to \(0\).

Solve using the zero-product property.

From \(x(x-7)=0\), the zero-product property gives \(x=0\) or \(x-7=0\), so \(x=7\).

What is the quadratic formula solution?

For \(x^2-7x+0=0\): \[x=\frac{7\pm\sqrt{49}}{2}=\frac{7\pm 7}{2}.\] So \(x=0\) or \(x=7\).

How can you interpret the solutions graphically?

Set \(y=x^2-7x=x(x-7)\). Zeros occur where the graph crosses the \(x\)-axis: at \(x=0\) and \(x=7\).

What is the vertex and does it affect the roots?

Vertex at \(x=\frac{-b}{2a}=\frac{7}{2}\). Roots remain solutions to \(x(x-7)=0\), so \(x=0,7\) regardless of the vertex position.

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