Derive the Product Form of the Quadratic f(x) = x² + x - 2

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

You need two numbers that multiply to give ac (the product of the coefficient of $ x^2 $ and the constant term) and add to give b (the coefficient of x). For $ x^2 + x - 2 $, find numbers that multiply to -2 and add to 1.

Q: Why do we rewrite the middle term?

Rewriting $ x $ as $ 2x - x $ lets us group terms in pairs that share common factors. This makes it easier to factor out common terms and find the final factored form.

Q: How can I check if my factored answer is correct?

Simply expand your factored form using FOIL or the distributive property. If you get back the original quadratic, your factoring is correct! For example: $ (x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2 $ ✓

Quadratic Factoring with Finding Factor Pairs

Find the representation of the product of the following function

$f(x)=x^2+x-2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Convert trinomial to multiplication

00:03 Identify the trinomial coefficients

00:07 We want to find 2 numbers whose sum equals coefficient B

00:12 and their product equals coefficient C

00:17 These are the matching numbers, let's substitute in multiplication

00:24 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

Find the representation of the product of the following function

$f(x)=x^2+x-2$

Step-by-step solution

To determine the product representation of $f(x) = x^2 + x - 2$ , we can factor the quadratic equation by following these steps:

Step 1: Identify the product $ac = 1 \times (-2) = -2$ and sum $b = 1$ .
Step 2: Find two numbers that multiply to $-2$ and add to $1$ . These numbers are $2$ and $-1$ .
Step 3: Rewrite the middle term using these numbers: $x^2 + 2x - 1x - 2$ .
Step 4: Factor by grouping:
- Group $x^2 + 2x$ and $-1x - 2$ as separate pairs:
- $x(x + 2) - 1(x + 2)$ .
Step 5: Factor out the common terms:
$(x + 2)(x - 1)$ .

Thus, the product representation of the function is $(x + 2)(x - 1)$ .

Final Answer

$(x+2)(x-1)$

Key Points to Remember

Essential concepts to master this topic

Product-Sum Method: Find two numbers that multiply to $ac$ and add to $b$
Factor by Grouping: Rewrite $x^2 + x - 2$ as $x^2 + 2x - x - 2$ then group pairs
Verify: Expand $(x+2)(x-1) = x^2 + x - 2$ to confirm factoring ✓

Common Mistakes

Avoid these frequent errors

Getting the signs wrong in the factored form
Don't just focus on finding numbers 2 and -1 without checking their placement = wrong signs like (x-2)(x+1)! The numbers 2 and -1 must be positioned so they create the correct middle term when expanded. Always verify by expanding your factored form back to the original.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=3,b=0,c=-3 $$

FAQ

Everything you need to know about this question

How do I know which two numbers to look for?

You need two numbers that multiply to give ac (the product of the coefficient of $x^2$ and the constant term) and add to give b (the coefficient of x). For $x^2 + x - 2$ , find numbers that multiply to -2 and add to 1.

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

Double-check your ac and b values first! If you still can't find suitable numbers, the quadratic might not factor nicely with integers, and you may need to use the quadratic formula instead.

Why do we rewrite the middle term?

Rewriting $x$ as $2x - x$ lets us group terms in pairs that share common factors. This makes it easier to factor out common terms and find the final factored form.

How can I check if my factored answer is correct?

Simply expand your factored form using FOIL or the distributive property. If you get back the original quadratic, your factoring is correct! For example: $(x+2)(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2$ ✓

What if the coefficient of x² isn't 1?

The same method works! Just remember that a is the coefficient of $x^2$ , so $ac$ will be different. For $2x^2 + 5x - 3$ , you'd look for numbers that multiply to $2 \times (-3) = -6$ and add to 5.

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Derive the Product Form of the Quadratic f(x) = x² + x - 2

Quadratic Factoring with Finding Factor Pairs

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

How do I know which two numbers to look for?

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

Why do we rewrite the middle term?

How can I check if my factored answer is correct?

What if the coefficient of x² isn't 1?

🌟 Unlock Your Math Potential

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