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 for X:

$x\cdot x=49$

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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