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

Q: How do I know which two numbers to look for?

For $ ax^2 + bx + c = 0 $, you need two numbers that multiply to give c and add to give b. In our case: multiply to 2, add to -3.

Q: What if I can't find two numbers that work?

If factoring doesn't work easily, try the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $. This always works for any quadratic equation!

Q: Why do I get x-1 and x-2 as factors instead of x+1 and x+2?

The numbers -1 and -2 become the opposite signs in the factors. If your numbers are -1 and -2, your factors are $ (x-1)(x-2) $.

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

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

Q: What if the coefficient of x² isn't 1?

When $ a \neq 1 $, factoring becomes trickier. Look for numbers that multiply to $ ac $ and add to $ b $, or use the quadratic formula for guaranteed results.

Quadratic Factoring with Integer Solutions

Solve the following equation:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the roots formula

00:23 Identify the coefficients

00:35 Substitute appropriate values according to the given data and solve

01:05 Calculate the square and products

01:17 One root is always equal to 1

01:27 These are the 2 possible solutions (addition,subtraction)

01:44 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:

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

Step-by-step solution

To solve the quadratic equation $x^2 - 3x + 2 = 0$ , we'll follow these steps:

Step 1: Check for factorization. Assuming the quadratic can be factored, we look for two numbers that multiply to $c = 2$ and add to $b = -3$ . These numbers are $-1$ and $-2$ .
Step 2: Factor the quadratic as $(x - 1)(x - 2) = 0$ .
Step 3: Solve each factor for $x$ .

Now, let's solve the factors:
From $(x - 1) = 0$ , we have $x = 1$ .
From $(x - 2) = 0$ , we have $x = 2$ .

Thus, the solutions to the equation are $x_1 = 1$ and $x_2 = 2$ .

Therefore, the solution to the problem is $x_1 = 1, x_2 = 2$ .

Final Answer

$x_1=1,x_2=2$

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 $x^2-3x+2$ , use -1 and -2: (-1)(-2)=2, (-1)+(-2)=-3
Check: Substitute x=1: $1^2-3(1)+2 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Finding numbers that add to c and multiply to b
Don't look for numbers that add to 2 and multiply to -3 = wrong factorization! This mixes up the roles of coefficients and leads to incorrect factors. Always find numbers that multiply to the constant term c and add to the coefficient b.

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

How do I know which two numbers to look for?

For $ax^2 + bx + c = 0$ , you need two numbers that multiply to give c and add to give b. In our case: multiply to 2, add to -3.

What if I can't find two numbers that work?

If factoring doesn't work easily, try the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . This always works for any quadratic equation!

Why do I get x-1 and x-2 as factors instead of x+1 and x+2?

The numbers -1 and -2 become the opposite signs in the factors. If your numbers are -1 and -2, your factors are $(x-1)(x-2)$ .

How can I check if my factoring is correct?

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

What if the coefficient of x² isn't 1?

When $a \neq 1$ , factoring becomes trickier. Look for numbers that multiply to $ac$ and add to $b$ , or use the quadratic formula for guaranteed results.

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