Solve the Quadratic Equation by Factoring: 3x² - 9x = 0

Q: Why do I need to factor out 3x first?

Factoring out the greatest common factor simplifies the problem! Both terms share $ 3x $, so pulling it out gives you $ 3x(x - 3) = 0 $, which is much easier to solve.

Q: What is the zero product property?

The zero product property says that if two factors multiply to zero, then at least one factor must equal zero. So if $ 3x(x - 3) = 0 $, then either $ 3x = 0 $ or $ x - 3 = 0 $.

Q: Why are there two solutions to this equation?

Quadratic equations can have up to two solutions because they involve $ x^2 $. In this case, both x = 0 and x = 3 make the original equation true when substituted back.

Q: What if I can't see the common factor right away?

Look at each term's coefficients and variables separately. Here, both $ 3x^2 $ and $ 9x $ contain 3 and x, so the GCF is $ 3x $.

Q: Can I solve this equation without factoring?

Yes, you could use the quadratic formula, but factoring is much faster here! Since there's no constant term, factoring out the GCF gives you the solutions immediately.

Quadratic Factoring with Zero Product Property

Solve the following equation:

$3x^2-9x=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Factor X squared into factors X and X

00:11 Factor 9 into factors 3 and 3

00:18 Find the common factor

00:34 Take out the common factor from the parentheses

00:40 Find what makes each factor in the product zero, so it equals 0

00:46 Isolate X, this is one solution

00:51 Now we'll use the same method and find the second solution

01:05 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$3x^2-9x=0$

Step-by-step solution

To solve the equation $3x^2 - 9x = 0$ , we will factor the quadratic expression:

Step 1: Factor the quadratic.
Observe that both terms in $3x^2 - 9x$ have a common factor of $3x$ . Thus, we factor by pulling out $3x$ :
$3x(x - 3) = 0$
Step 2: Apply the zero product property.
According to the zero product property, if $3x(x - 3) = 0$ , then either $3x = 0$ or $x - 3 = 0$ must hold true.
Step 3: Solve each factor for $x$ .
- For $3x = 0$ :
Divide both sides by 3:
$x = 0$
- For $x - 3 = 0$ :
Add 3 to both sides:
$x = 3$

Therefore, the solutions to the equation are $x_1 = 3$ and $x_2 = 0$ .

These solutions correspond to the choice: $x_1=3,x_2=0$ .

Final Answer

$x_1=3,x_2=0$

Key Points to Remember

Essential concepts to master this topic

Factoring: Always pull out the greatest common factor first
Technique: Factor $3x^2 - 9x$ as $3x(x - 3) = 0$
Check: Substitute both solutions: $3(0)^2 - 9(0) = 0$ and $3(3)^2 - 9(3) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to factor out the GCF first
Don't try to factor $x^2 - 3x$ directly without pulling out 3x first = missed solutions! This makes factoring much harder and often leads to incomplete answers. Always look for and factor out the greatest common factor as your first step.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why do I need to factor out 3x first?

Factoring out the greatest common factor simplifies the problem! Both terms share $3x$ , so pulling it out gives you $3x(x - 3) = 0$ , which is much easier to solve.

What is the zero product property?

The zero product property says that if two factors multiply to zero, then at least one factor must equal zero. So if $3x(x - 3) = 0$ , then either $3x = 0$ or $x - 3 = 0$ .

Why are there two solutions to this equation?

Quadratic equations can have up to two solutions because they involve $x^2$ . In this case, both x = 0 and x = 3 make the original equation true when substituted back.

What if I can't see the common factor right away?

Look at each term's coefficients and variables separately. Here, both $3x^2$ and $9x$ contain 3 and x, so the GCF is $3x$ .

Can I solve this equation without factoring?

Yes, you could use the quadratic formula, but factoring is much faster here! Since there's no constant term, factoring out the GCF gives you the solutions immediately.

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