Quadratic Equation Basics: Solving x² + x = 0

Q: Why can't I just divide both sides by x to get x + 1 = 0?

Never divide by a variable that could equal zero! If x = 0, you'd be dividing by zero, which is undefined. You'd also lose the solution x = 0 completely. Always factor instead.

Q: How do I know when to use factoring vs. the quadratic formula?

Start with factoring! Look for common factors first, then try factoring patterns. Only use the quadratic formula when factoring doesn't work easily.

Q: What is the zero product property exactly?

If two things multiply to give zero, then at least one of them must be zero. So if $ x(x + 1) = 0 $, then either x = 0 or (x + 1) = 0.

Q: Why do I get two answers for one equation?

Quadratic equations can have up to 2 solutions! This is normal and expected. Both solutions are correct - you can verify by substituting each one back into the original equation.

Q: What if I can't factor easily?

Then it's time for the quadratic formula! It works for any quadratic equation: $ x = \frac{-b ± \sqrt{b^2-4ac}}{2a} $ where your equation is in the form $ ax^2 + bx + c = 0 $.

Quadratic Equations with Common Factor Method

Solve the following equation:

$x^2+x=0$

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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

00:09 Find the common factor

00:21 Take out the common factor from the parentheses

00:25 Find what makes each factor in the product equal to 0

00:31 This is one solution

00:36 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:

$x^2+x=0$

Step-by-step solution

To solve the equation $x^2 + x = 0$ , we will use a step-by-step method:

Step 1: Identify the equation form.
The equation given is $x^2 + x = 0$ . This is a quadratic equation in a simpler form since it can be factored easily.
Step 2: Factor the equation.
We notice that both terms of the equation have a common factor, which is $x$ . Therefore, we can factor out $x$ as follows:

$x(x + 1) = 0$

Step 3: Apply the zero product property.
The zero product property tells us that if the product of two factors is zero, then at least one of the factors must be zero. Thus, set each factor equal to zero:

$x = 0$
$x + 1 = 0$

Step 4: Solve each equation.
The first equation gives $x = 0$ directly. The second equation can be solved by subtracting 1 from both sides to find:

$x = -1$

Therefore, the solutions to the equation $x^2 + x = 0$ are $x_1 = 0$ and $x_2 = -1$ .

Therefore, the correct answer is:

$x_1 = 0, x_2 = -1$

Final Answer

$x_1=0,x_2=-1$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Look for common factors before using quadratic formula
Technique: Factor out x from $x^2 + x = 0$ to get $x(x + 1) = 0$
Check: Substitute both solutions: $0^2 + 0 = 0$ and $(-1)^2 + (-1) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Using the quadratic formula unnecessarily
Don't jump straight to $x = \frac{-b ± \sqrt{b^2-4ac}}{2a}$ when you can factor easily = makes simple problems complicated! You'll waste time and risk calculation errors. Always check for common factors first before using the quadratic formula.

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 can't I just divide both sides by x to get x + 1 = 0?

Never divide by a variable that could equal zero! If x = 0, you'd be dividing by zero, which is undefined. You'd also lose the solution x = 0 completely. Always factor instead.

How do I know when to use factoring vs. the quadratic formula?

Start with factoring! Look for common factors first, then try factoring patterns. Only use the quadratic formula when factoring doesn't work easily.

What is the zero product property exactly?

If two things multiply to give zero, then at least one of them must be zero. So if $x(x + 1) = 0$ , then either x = 0 or (x + 1) = 0.

Why do I get two answers for one equation?

Quadratic equations can have up to 2 solutions! This is normal and expected. Both solutions are correct - you can verify by substituting each one back into the original equation.

Can a quadratic equation have just one solution?

Yes! Sometimes both solutions are the same number (called a repeated root), or the parabola just touches the x-axis at one point instead of crossing it twice.

What if I can't factor easily?

Then it's time for the quadratic formula! It works for any quadratic equation: $x = \frac{-b ± \sqrt{b^2-4ac}}{2a}$ where your equation is in the form $ax^2 + bx + c = 0$ .

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