Solving the Quadratic Equation: -2x² + 4x = 0

Q: Why can't I just divide both sides by x to simplify?

Never divide by a variable! When you divide by x, you might be dividing by zero, which eliminates valid solutions. In this problem, x = 0 is one of our answers, so dividing by x would make us lose it.

Q: How do I know what to factor out?

Look for the greatest common factor (GCF) of all terms. Here, both $ -2x^2 $ and $ 4x $ have 2x in common, so factor out 2x first.

Q: What if I factored it differently?

You might factor as $ -2x(x - 2) = 0 $ instead. That's fine! You'll still get the same solutions: x = 0 and x = 2. Different factoring forms can lead to the same answer.

Q: Why does setting each factor to zero work?

This uses the Zero Product Property: if two things multiply to give zero, then at least one of them must be zero. So if $ 2x(-x + 2) = 0 $, then either 2x = 0 or (-x + 2) = 0.

Q: How can I check if my factoring is correct?

Expand your factored form back out! $ 2x(-x + 2) = -2x^2 + 4x $, which matches our original equation. If it doesn't match, redo the factoring.

Quadratic Equations with Factoring Method

Solve the following equation:

$-2x^2+4x=0$

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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

00:10 Factor 4 into factors 2 and 2

00:13 Find the largest common factor

00:34 Take out this factor from the parentheses

00:38 Find what makes each factor equal to zero in the multiplication

00:51 Isolate X, this is one solution

00:58 Find what makes the second factor equal to zero

01:02 Isolate X

01:08 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$-2x^2+4x=0$

Step-by-step solution

To solve this quadratic equation, we will use factoring.

Consider the given equation:
$-2x^2 + 4x = 0$

Step 1: Factor out the greatest common factor from the terms.

The common factor of $-2x^2$ and $4x$ is $2x$ . Factoring this out gives:
$2x(-x + 2) = 0$

Step 2: Set each factor to zero.

$2x = 0$
$-x + 2 = 0$

Step 3: Solve each equation.

For $2x = 0$ :
Divide both sides by 2, resulting in: $x = 0$ .
For $-x + 2 = 0$ :
Add $x$ to both sides, giving $2 = x$ or $x = 2$ .

Therefore, the solutions to the equation are $x_1 = 2$ and $x_2 = 0$ .

The correct answer is choice 4: $x_1 = 2, x_2 = 0$ .

Final Answer

$x_1=2,x_2=0$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find greatest common factor first before solving
Technique: Factor out 2x from $-2x^2 + 4x = 2x(-x + 2)$
Check: Substitute x = 0 and x = 2: both give 0 = 0 ✓

Common Mistakes

Avoid these frequent errors

Dividing both sides by x first
Don't divide by x immediately to get -2x + 4 = 0! This loses the solution x = 0 because you're dividing by zero. Always factor out the common term first, then set each factor equal to zero.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

Why can't I just divide both sides by x to simplify?

Never divide by a variable! When you divide by x, you might be dividing by zero, which eliminates valid solutions. In this problem, x = 0 is one of our answers, so dividing by x would make us lose it.

How do I know what to factor out?

Look for the greatest common factor (GCF) of all terms. Here, both $-2x^2$ and $4x$ have 2x in common, so factor out 2x first.

What if I factored it differently?

You might factor as $-2x(x - 2) = 0$ instead. That's fine! You'll still get the same solutions: x = 0 and x = 2. Different factoring forms can lead to the same answer.

Why does setting each factor to zero work?

This uses the Zero Product Property: if two things multiply to give zero, then at least one of them must be zero. So if $2x(-x + 2) = 0$ , then either 2x = 0 or (-x + 2) = 0.

How can I check if my factoring is correct?

Expand your factored form back out! $2x(-x + 2) = -2x^2 + 4x$ , which matches our original equation. If it doesn't match, redo the factoring.

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