Find the Standard Representation of (2x+1)² - 1

Q: Why can't I just expand (2x+1)² and call it done?

Because the original function is $ (2x+1)^2 - 1 $, not just $ (2x+1)^2 $! You must subtract 1 from the expanded form to get the complete standard representation.

Q: How do I remember the binomial square formula?

Think of it as (First + Last)² = First² + 2(First)(Last) + Last². For $ (2x+1)^2 $: First² = $ (2x)^2 = 4x^2 $, 2(First)(Last) = $ 2(2x)(1) = 4x $, Last² = $ 1^2 = 1 $.

Q: What if I get confused with the signs?

Take it step by step! First expand the square completely, then handle the subtraction. Write it as: $ (4x^2 + 4x + 1) - 1 $ to clearly see that you're subtracting 1 from the constant term.

Q: How can I check if my standard form is correct?

Substitute a simple value like $ x = 1 $ into both forms. Original: $ (2(1)+1)^2 - 1 = 3^2 - 1 = 8 $. Standard form: $ 4(1)^2 + 4(1) = 8 $. If they match, you're correct!

Q: Why is this called 'standard form'?

Standard form for quadratic functions is $ ax^2 + bx + c $ where terms are arranged by descending powers. This makes it easy to identify coefficients and work with the function.

Quadratic Expansion with Binomial Squares

Find the standard representation of the following function

$f(x)=(2x+1)^2-1$

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Step-by-step video solution

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00:00 Simplify to the standard representation of the function

00:04 Expand parentheses according to the shortened multiplication formulas

00:16 Calculate powers and products

00:25 Collect terms

00:30 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 standard representation of the following function

$f(x)=(2x+1)^2-1$

Step-by-step solution

To convert $f(x) = (2x+1)^2 - 1$ into its standard quadratic form, we need to expand $(2x+1)^2$ first and then adjust for the subtraction of 1.

The expansion is carried out using the binomial expansion formula:

$(2x + 1)^2 = (2x)^2 + 2(2x)(1) + 1^2$ .

Calculating each term gives:

$(2x)^2 = 4x^2$
$2(2x)(1) = 4x$
$1^2 = 1$

Combining these, we obtain:

$(2x + 1)^2 = 4x^2 + 4x + 1$

Now, substituting back into the original equation:

$f(x) = (2x+1)^2 - 1 = (4x^2 + 4x + 1) - 1$

Subtracting 1 from the constant term, we get:

$f(x) = 4x^2 + 4x + 1 - 1 = 4x^2 + 4x$

Therefore, the standard form representation of the function is $f(x) = 4x^2 + 4x$ .

Final Answer

$f(x)=4x^2+4x$

Key Points to Remember

Essential concepts to master this topic

Formula: Use (a+b)² = a² + 2ab + b² for perfect squares
Technique: Expand (2x+1)² = 4x² + 4x + 1, then subtract 1
Check: Standard form has descending powers: 4x² + 4x + 0 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to subtract 1 from the expanded form
Don't expand (2x+1)² = 4x² + 4x + 1 and stop there = wrong answer 4x² + 4x + 1! You must subtract the final -1 from the original expression. Always complete all operations: (4x² + 4x + 1) - 1 = 4x² + 4x.

Practice Quiz

Test your knowledge with interactive questions

Create an algebraic expression based on the following parameters:

$$ a=2,b=2,c=2 $$

FAQ

Everything you need to know about this question

Why can't I just expand (2x+1)² and call it done?

Because the original function is $(2x+1)^2 - 1$ , not just $(2x+1)^2$ ! You must subtract 1 from the expanded form to get the complete standard representation.

How do I remember the binomial square formula?

Think of it as (First + Last)² = First² + 2(First)(Last) + Last². For $(2x+1)^2$ : First² = $(2x)^2 = 4x^2$ , 2(First)(Last) = $2(2x)(1) = 4x$ , Last² = $1^2 = 1$ .

What if I get confused with the signs?

Take it step by step! First expand the square completely, then handle the subtraction. Write it as: $(4x^2 + 4x + 1) - 1$ to clearly see that you're subtracting 1 from the constant term.

How can I check if my standard form is correct?

Substitute a simple value like $x = 1$ into both forms. Original: $(2(1)+1)^2 - 1 = 3^2 - 1 = 8$ . Standard form: $4(1)^2 + 4(1) = 8$ . If they match, you're correct!

Why is this called 'standard form'?

Standard form for quadratic functions is $ax^2 + bx + c$ where terms are arranged by descending powers. This makes it easy to identify coefficients and work with the function.

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