Transforming the Function: Reveal the Standard Form of f(x) = (x+5)² + 3

Q: Why can't I just distribute the square to each term inside?

Because (x+5)² is NOT equal to x² + 5²! The square applies to the entire binomial. You must use the binomial expansion formula: $ (a+b)^2 = a^2 + 2ab + b^2 $.

Q: What's the difference between vertex form and standard form?

Vertex form is $ f(x) = a(x-h)^2 + k $ which shows the vertex clearly. Standard form is $ f(x) = ax^2 + bx + c $ which is fully expanded and easier for some calculations.

Q: How do I remember the binomial expansion formula?

Think "First, Outer, Inner, Last" (FOIL): $ (x+5)^2 = (x+5)(x+5) $. First: x·x = x². Outer: x·5 = 5x. Inner: 5·x = 5x. Last: 5·5 = 25. Combined: x² + 5x + 5x + 25 = x² + 10x + 25.

Q: Can I check my answer by plugging in a number?

Absolutely! Pick any value like x = 1. Original: $ (1+5)^2 + 3 = 36 + 3 = 39 $. Standard form: $ 1^2 + 10(1) + 28 = 1 + 10 + 28 = 39 $. Same result means you're correct!

Q: What if I get confused with the signs?

Be extra careful with positive signs! Since we have (x+5)², both terms inside are positive. When expanding, you get +10x (not -10x) and +25. Then add the +3 to get +28 as your final constant term.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify to the standard representation of the function

00:03 Open parentheses according to the shortened multiplication formulas

00:11 Calculate powers and products

00:28 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

$f(x)=(x+5)^2+3$

2

Step-by-step solution

To convert the given quadratic function into its standard form, follow these steps:

Step 1: Expand the Binomial
We begin with the function in vertex form: $f(x) = (x + 5)^2 + 3$ . The expression $(x + 5)^2$ can be expanded using the binomial theorem: $(a + b)^2 = a^2 + 2ab + b^2$ .
Step 2: Apply the Expansion Formula
Let $a = x$ and $b = 5$ . Therefore, $(x + 5)^2 = x^2 + 2 \times x \times 5 + 5^2 = x^2 + 10x + 25$ .
Step 3: Add the Constant
Now, add the constant 3 to this expanded result: $x^2 + 10x + 25 + 3 = x^2 + 10x + 28$ .

Thus, the standard representation of the function is $f(x) = x^2 + 10x + 28$ .

Given the choices, the correct answer is $f(x) = x^2 + 10x + 28$ , which matches choice 2.

3

Final Answer

$f(x)=x^2+10x+28$

Key Points to Remember

Essential concepts to master this topic

Binomial Rule: $(a+b)^2 = a^2 + 2ab + b^2$ for perfect square trinomials
Technique: Expand $(x+5)^2 = x^2 + 10x + 25$ then add constant
Check: Verify by substituting a value: if x=0, both forms give f(0)=28 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to add the constant term outside the parentheses
Don't expand (x+5)² = x² + 10x + 25 and forget the +3! This gives f(x) = x² + 10x + 25 instead of the correct x² + 10x + 28. The +3 is separate from the squared term and must be added to get the final constant. Always add all terms together after expanding.

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 distribute the square to each term inside?

+

Because (x+5)² is NOT equal to x² + 5²! The square applies to the entire binomial. You must use the binomial expansion formula: $(a+b)^2 = a^2 + 2ab + b^2$ .

What's the difference between vertex form and standard form?

+

Vertex form is $f(x) = a(x-h)^2 + k$ which shows the vertex clearly. Standard form is $f(x) = ax^2 + bx + c$ which is fully expanded and easier for some calculations.

How do I remember the binomial expansion formula?

+

Think "First, Outer, Inner, Last" (FOIL): $(x+5)^2 = (x+5)(x+5)$ . First: x·x = x². Outer: x·5 = 5x. Inner: 5·x = 5x. Last: 5·5 = 25. Combined: x² + 5x + 5x + 25 = x² + 10x + 25.

Can I check my answer by plugging in a number?

+

Absolutely! Pick any value like x = 1. Original: $(1+5)^2 + 3 = 36 + 3 = 39$ . Standard form: $1^2 + 10(1) + 28 = 1 + 10 + 28 = 39$ . Same result means you're correct!

What if I get confused with the signs?

+

Be extra careful with positive signs! Since we have (x+5)², both terms inside are positive. When expanding, you get +10x (not -10x) and +25. Then add the +3 to get +28 as your final constant term.

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Transforming the Function: Reveal the Standard Form of f(x) = (x+5)² + 3

Quadratic Expansion with Vertex Form

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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

Why can't I just distribute the square to each term inside?

What's the difference between vertex form and standard form?

How do I remember the binomial expansion formula?

Can I check my answer by plugging in a number?

What if I get confused with the signs?

🌟 Unlock Your Math Potential

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