Solve the Quadratic Equation: x² - 2x - 3 = 0

Q: How do I find the two numbers that work for factoring?

Look for two numbers that multiply to give the constant term (here -3) and add to give the middle coefficient (here -2). Try factor pairs of -3: (-3,+1) and (+3,-1). Check: -3×1 = -3 ✓ and -3+1 = -2 ✓

Q: What if I can't factor the quadratic easily?

Not all quadratics factor nicely! You can always use the quadratic formula as a backup method: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $

Q: Why do I get two different answers?

Quadratic equations usually have two solutions because they represent parabolas that cross the x-axis at two points. Both x = 3 and x = -1 make the original equation true!

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

Expand your factored form back out! $ (x-3)(x+1) = x^2 + x - 3x - 3 = x^2 - 2x - 3 $. If you get the original equation, your factoring is correct.

Q: What does it mean when the equation equals zero?

Setting the quadratic equal to zero helps us find the x-intercepts or roots of the parabola. These are the points where the graph crosses the x-axis.

Quadratic Equations with Factoring Method

$x^2-2x-3=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 We'll break it down using trinomials, let's look at the coefficients

00:08 We want to find 2 numbers whose sum equals B (-2)

00:16 and their product equals C (-3)

00:22 These are the matching numbers, let's substitute in parentheses

00:33 Let's find what zeros each factor

00:43 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

$x^2-2x-3=0$

Step-by-step solution

Let's observe that the given equation:

$x^2-2x-3=0$ is a quadratic equation that can be solved using quick factoring:

$x^2-2x-3=0 \longleftrightarrow\begin{cases}\underline{?}\cdot\underline{?}=-3\\ \underline{?}+\underline{?}=-2\end{cases}\\ \downarrow\\ (x-3)(x+1)=0$ and therefore we get two simpler equations from which we can extract the solution:

$(x-3)(x+1)=0 \\ \downarrow\\ x-3=0\rightarrow\boxed{x=3}\\ x+1=0\rightarrow\boxed{x=-1}\\ \boxed{x=-1,3}$ Therefore, the correct answer is answer B.

Final Answer

$x=3,x=-1$

Key Points to Remember

Essential concepts to master this topic

Factoring Rule: Find two numbers that multiply to c and add to b
Technique: For $x^2-2x-3$ , need numbers multiplying to -3, adding to -2
Check: Substitute both solutions: $3^2-2(3)-3=0$ and $(-1)^2-2(-1)-3=0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing the signs when factoring
Don't write (x+3)(x-1) when you need numbers that add to -2! This gives +2 instead of -2. The factors -3 and +1 multiply to -3 and add to -2. Always check that your factors multiply to c AND add to b coefficient.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

How do I find the two numbers that work for factoring?

Look for two numbers that multiply to give the constant term (here -3) and add to give the middle coefficient (here -2). Try factor pairs of -3: (-3,+1) and (+3,-1). Check: -3×1 = -3 ✓ and -3+1 = -2 ✓

What if I can't factor the quadratic easily?

Not all quadratics factor nicely! You can always use the quadratic formula as a backup method: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Why do I get two different answers?

Quadratic equations usually have two solutions because they represent parabolas that cross the x-axis at two points. Both x = 3 and x = -1 make the original equation true!

How can I check if my factoring is correct?

Expand your factored form back out! $(x-3)(x+1) = x^2 + x - 3x - 3 = x^2 - 2x - 3$ . If you get the original equation, your factoring is correct.

What does it mean when the equation equals zero?

Setting the quadratic equal to zero helps us find the x-intercepts or roots of the parabola. These are the points where the graph crosses the x-axis.

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