Convert to Standard Form: Unfolding f(x) = -(x+1)(x-1)

Quadratic Functions with Difference of Squares

Find the standard representation of the following function

f(x)=−(x+1)(x−1) f(x)=-(x+1)(x-1)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Simplify to the standard representation of the function
00:05 Open parentheses according to shortened multiplication formulas
00:28 Negative multiplied by positive is always negative
00:31 Negative multiplied by negative is always positive
00:34 And this is the solution to the question

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+1)(x−1) f(x)=-(x+1)(x-1)

2

Step-by-step solution

To solve this problem, we need to convert the given function f(x)=−(x+1)(x−1) f(x) = -(x+1)(x-1) from its current product form to standard form.

Let's follow these steps:

  • Step 1: Recognize that the expression (x+1)(x−1) (x+1)(x-1) is a standard difference of squares formula, expressed as (a+b)(a−b)=a2−b2 (a+b)(a-b) = a^2 - b^2 , where a=x a = x and b=1 b = 1 .
  • Step 2: According to the formula, (x+1)(x−1)=x2−12=x2−1 (x+1)(x-1) = x^2 - 1^2 = x^2 - 1 .
  • Step 3: Substitute this result back into the function: f(x)=−(x2−1) f(x) = -(x^2 - 1) .
  • Step 4: Simplify the expression by distributing the negative sign: −(x2−1)=−x2+1 -(x^2 - 1) = -x^2 + 1 .

Therefore, the function in its standard form is f(x)=−x2+1 f(x) = -x^2 + 1 .

This matches with choice 1: f(x)=−x2+1 f(x)=-x^2+1 .

3

Final Answer

f(x)=−x2+1 f(x)=-x^2+1

Key Points to Remember

Essential concepts to master this topic
  • Pattern Recognition: Identify (a+b)(a-b) = a² - b² formula
  • Technique: (x+1)(x-1) becomes x² - 1, then apply negative sign
  • Check: Expand -x² + 1 back to -(x+1)(x-1) to verify ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute the negative sign
    Don't just expand (x+1)(x-1) = x² - 1 and stop there! This ignores the negative sign outside the parentheses, giving x² - 1 instead of -x² + 1. Always distribute the negative sign to every term: -(x² - 1) = -x² + 1.

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 is (x+1)(x-1) a difference of squares?

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Because it follows the pattern (a+b)(a-b) where a = x and b = 1. This special pattern always equals a2−b2 a^2 - b^2 , so (x+1)(x−1)=x2−12=x2−1 (x+1)(x-1) = x^2 - 1^2 = x^2 - 1 .

What happens to the negative sign in front?

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The negative sign distributes to every term inside the parentheses. So −(x2−1) -(x^2 - 1) becomes −x2+1 -x^2 + 1 . Remember: negative times negative equals positive!

How do I know when to use the difference of squares formula?

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Look for the pattern (something + number)(something - same number). Examples: (x+3)(x-3), (2y+5)(2y-5), or (a+1)(a-1). The middle terms always cancel out!

Can I just multiply it out the long way instead?

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Yes, but recognizing the difference of squares pattern is much faster! Plus, it helps you see the structure of quadratic expressions better. Both methods give the same answer.

What if I got -x² - 1 as my answer?

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You probably forgot to distribute the negative sign correctly. Remember: −(x2−1)=−x2−(−1)=−x2+1 -(x^2 - 1) = -x^2 - (-1) = -x^2 + 1 . The double negative becomes positive!

Is this function a parabola?

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Yes! f(x)=−x2+1 f(x) = -x^2 + 1 is a parabola opening downward because of the negative coefficient on x². The vertex is at (0, 1), which is the highest point.

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