Solve the Quadratic Equation: 5x² + 10x = 0 Using Factoring

Q: Why do I factor out 5x instead of just x?

You want the greatest common factor (GCF) to simplify as much as possible! Both terms $ 5x^2 $ and $ 10x $ contain 5x, so factoring out 5x gives you the simplest form.

Q: What if one of my solutions is zero?

Zero is a perfectly valid solution! In fact, many quadratic equations have x = 0 as one solution. Just make sure to find the other solution too using the zero product property.

Q: Can I solve this using the quadratic formula instead?

Yes, but factoring is much faster when you can easily identify common factors! The quadratic formula works for all quadratics, but factoring saves time when the equation has obvious factors like this one.

Q: What if I can't factor out anything obvious?

First, always check for a common factor in all terms. If there isn't one, try other factoring methods like grouping or factoring trinomials. If factoring seems impossible, use the quadratic formula.

Q: Do I always get two solutions for quadratic equations?

Most quadratic equations have two solutions, but sometimes both solutions are the same number (called a repeated root). Occasionally, there might be no real solutions at all!

Quadratic Factoring with Common Factors

Solve the following equation:

$5x^2+10x=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:10 Factor X squared into factors X and X

00:13 Factor 10 into factors 5 and 2

00:19 Find the common factor

00:34 Take out the common factor from the parentheses

00:41 Find what makes each factor in the product zero, so it equals 0

00:47 Isolate X, this is one solution

00:56 Now let's use the same method and find the second solution

01:03 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:

$5x^2+10x=0$

Step-by-step solution

To solve the quadratic equation $5x^2 + 10x = 0$ , we will follow these steps:

Step 1: Factor the equation: Identify the greatest common factor, which in this case is $5x$ .
Step 2: Apply the zero product property: Set each factor equal to zero and solve for $x$ .

Now, let's execute these steps:

Step 1: Factor the equation.
The equation is $5x^2 + 10x = 0$ .
Factor out the greatest common factor, $5x$ , from both terms:
$5x(x + 2) = 0$ .

Step 2: Apply the zero product property.
According to this property, if the product of factors is zero, then at least one of the factors must be zero.
Thus, we set each factor equal to zero:

$5x = 0$
$x + 2 = 0$

Solving these, we find:

$5x = 0 \Rightarrow x = 0$

$x + 2 = 0 \Rightarrow x = -2$

Therefore, the solutions to the equation $5x^2 + 10x = 0$ are $x_1 = 0$ and $x_2 = -2$ .

The correct choice among the provided options is: $x_1 = 0, x_2 = -2$ .

Final Answer

$x_1=0,x_2=-2$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Always extract the greatest common factor first
Technique: Factor out 5x: $5x^2 + 10x = 5x(x + 2)$
Check: Substitute x = 0 and x = -2 back into original equation ✓

Common Mistakes

Avoid these frequent errors

Forgetting to apply zero product property to both factors
Don't just solve 5x = 0 and ignore (x + 2) = 0! This gives you only one solution instead of two. The zero product property requires BOTH factors to be set equal to zero. Always solve 5x = 0 AND x + 2 = 0 to find all solutions.

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 factor out 5x instead of just x?

You want the greatest common factor (GCF) to simplify as much as possible! Both terms $5x^2$ and $10x$ contain 5x, so factoring out 5x gives you the simplest form.

What if one of my solutions is zero?

Zero is a perfectly valid solution! In fact, many quadratic equations have x = 0 as one solution. Just make sure to find the other solution too using the zero product property.

Can I solve this using the quadratic formula instead?

Yes, but factoring is much faster when you can easily identify common factors! The quadratic formula works for all quadratics, but factoring saves time when the equation has obvious factors like this one.

How do I know when to use the zero product property?

Use the zero product property whenever you have an equation in the form (something)(something) = 0. If the product equals zero, then at least one of the factors must equal zero!

What if I can't factor out anything obvious?

First, always check for a common factor in all terms. If there isn't one, try other factoring methods like grouping or factoring trinomials. If factoring seems impossible, use the quadratic formula.

Do I always get two solutions for quadratic equations?

Most quadratic equations have two solutions, but sometimes both solutions are the same number (called a repeated root). Occasionally, there might be no real solutions at all!

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