Solve the equation by factoring.
3​x(xminus3​)equals10xsquaredminus11x
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Part 1
Rewrite the equation in factored form.
  
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​(Factor completely.)
write the equation in factored form and give the solution set

To solve the equation by factoring, let's start by rewriting the given equation in a standard quadratic form:

Given:
\[
3x(x - 3) = 10x^2 - 11x
\]

### Step 1: Expand and rearrange the equation

First, expand the left side:

\[
3x(x - 3) = 3x^2 - 9x
\]

Now, rewrite the equation with all terms on one side:

\[
3x^2 - 9x - (10x^2 - 11x) = 0
\]

Simplify the equation:

\[
3x^2 - 9x - 10x^2 + 11x = 0
\]

Combine like terms:

\[
-7x^2 + 2x = 0
\]

### Step 2: Factor the equation

Factor out the greatest common factor, which is \(x\):

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

### Step 3: Find the solutions

Set each factor equal to zero:

1. \(x = 0\)
2. \(-7x + 2 = 0\)

Solve for \(x\) in the second equation:

\[
-7x + 2 = 0 \implies -7x = -2 \implies x = \frac{2}{7}
\]

### Solution Set

The solution set is:

\[
\{0, \frac{2}{7}\}
\]

### Final Factored Form

The equation in completely factored form is:

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

Would you like more details on the factoring process, or do you have any other questions?

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Here are some related questions to expand your understanding:

1. How do you solve quadratic equations by completing the square?
2. What are the methods to solve quadratic equations other than factoring?
3. How do you find the vertex of a parabola given in standard form?
4. Can a quadratic equation have complex solutions, and how are they found?
5. What is the quadratic formula and when is it most useful?

**Tip:** When solving quadratic equations, always check if factoring is possible before using other methods like the quadratic formula.

Solve the equation by factoring. 3x(x - 3) = 10x^2 - 11x. Rewrite the equation in factored form and give the solution set.

Learn how to solve the quadratic equation 3x(x - 3) = 10x^2 - 11x by factoring. This step-by-step guide covers expanding the equation, finding the greatest common factor, and applying the zero product property to find the solution set. Suitable for students in Grades 9-10.

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