Standard Representation: Simplify f(x) = (-x+2)(x+3)

Quadratic Functions with Binomial Expansion

Find the standard representation of the following function

f(x)=(x+2)(x+3) f(x)=(-x+2)(x+3)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:08 Let's start by representing the function in a simple standard form.
00:13 Make sure to open the parentheses correctly. Multiply each term in the first set, by each term in the second set. Take your time with this step.
00:25 Now, let's gather like terms together to simplify.
00:29 And that's how we solve the problem! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find the standard representation of the following function

f(x)=(x+2)(x+3) f(x)=(-x+2)(x+3)

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Use the FOIL method to expand the product of binomials.
  • Step 2: Combine like terms to simplify the expression.

Now, let's work through each step:

Step 1: Expand the product (x+2)(x+3)(-x+2)(x+3) using the FOIL method:
First terms: xx=x2-x \cdot x = -x^2
Outer terms: x3=3x-x \cdot 3 = -3x
Inner terms: 2x=2x2 \cdot x = 2x
Last terms: 23=62 \cdot 3 = 6

This gives us the expression:
x23x+2x+6 -x^2 - 3x + 2x + 6

Step 2: Combine like terms:
Combine 3x-3x and 2x2x to get x-x.
Thus, the expression simplifies to:
x2x+6 -x^2 - x + 6

Therefore, the standard form of the function is f(x)=x2x+6 f(x) = -x^2 - x + 6 .

3

Final Answer

f(x)=x2x+6 f(x)=-x^2-x+6

Key Points to Remember

Essential concepts to master this topic
  • FOIL Method: First, Outer, Inner, Last terms multiply systematically
  • Technique: (-x)(x) = -x², (-x)(3) = -3x, (2)(x) = 2x, (2)(3) = 6
  • Check: Substitute x = 0: f(0) = (2)(3) = 6 = -0² - 0 + 6 ✓

Common Mistakes

Avoid these frequent errors
  • Sign errors when multiplying negative terms
    Don't forget the negative sign in (-x+2) when multiplying = wrong signs throughout! Students often write (-x)(x) = x² instead of -x², making the final answer positive instead of negative. Always track negative signs carefully through each multiplication step.

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

Why do I keep getting the wrong signs in my answer?

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Sign errors are super common! Remember that (-x) × (x) = -x², not +x². Write out each multiplication step clearly and track negative signs through the entire process.

What's the difference between FOIL and distributive property?

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FOIL is just a memory tool for the distributive property with binomials! It helps you remember to multiply every term in the first binomial by every term in the second.

How do I know which terms to combine?

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Look for like terms - terms with the same variable and exponent. In this problem, -3x and +2x are like terms because they both have x¹.

Can I expand in a different order?

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Yes! You could distribute (-x+2) to each term in (x+3), or vice versa. The final answer will be the same as long as you don't miss any terms.

What if I get confused with the order of terms?

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Standard form for quadratic functions is ax² + bx + c. Always arrange your final answer with the highest degree term first: x² term, then x term, then constant.

How can I check if my expansion is correct?

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Pick a simple value like x = 1 and substitute into both the original and expanded forms. If f(1) gives the same result in both forms, your expansion is correct!

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