Transforming Quadratics: Find the Standard Form of f(x) = (x-3)² + x

Q: Why can't I just multiply (x-3)(x-3) instead of using the formula?

You absolutely can! Both methods work perfectly. FOIL method: (x-3)(x-3) = x² - 3x - 3x + 9 = x² - 6x + 9. The formula (a-b)² = a² - 2ab + b² is just faster once you memorize it.

Q: What does 'standard form' actually mean?

Standard form for a quadratic function is $ f(x) = ax^2 + bx + c $ where terms are arranged from highest to lowest degree. No parentheses, no factored terms - just expanded and simplified!

Q: How do I remember the (a-b)² formula?

Think of it as 'square the first, minus twice the product, plus square the last'. Or remember: (a-b)² = a² - 2ab + b². Practice with simple numbers like (x-2)² first!

Q: Can I leave my answer as (x-3)² + x?

No, that's vertex form, not standard form. The question specifically asks for standard form, which must be fully expanded as $ ax^2 + bx + c $.

Quadratic Expansion with Linear Terms

Find the standard representation of the following function:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify it to the standard form of the function.

00:12 Now, we will expand the brackets using multiplication formulas.

00:20 Next, let's calculate the powers and products.

00:29 Then, we'll collect all the like terms.

00:38 And there you have it! That's the solution.

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:

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

Step-by-step solution

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

Expand $(x-3)^2$ using the formula for a square:
$(x-3)^2 = x^2 - 2 \times x \times 3 + 3^2 = x^2 - 6x + 9$
Add the $x$ from $f(x) = (x-3)^2 + x$ to the expanded terms:
$x^2 - 6x + 9 + x$
Combine like terms:
$x^2 - 6x + x + 9 = x^2 - 5x + 9$

Therefore, the standard form of the function $f(x)$ is $f(x) = x^2 - 5x + 9$ .

Thus, the correct choice is Choice 3.

Final Answer

$f(x)=x^2-5x+9$

Key Points to Remember

Essential concepts to master this topic

Formula: Expand (a-b)² = a² - 2ab + b²
Technique: (x-3)² becomes x² - 6x + 9, then add x
Check: Substitute x=0: (0-3)² + 0 = 9, and 0² - 5(0) + 9 = 9 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to add the extra x term after expanding
Don't expand (x-3)² = x² - 6x + 9 and forget the +x = wrong final answer! Many students focus only on the squared term and miss the linear term. Always identify all terms in the original function before combining.

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 can't I just multiply (x-3)(x-3) instead of using the formula?

You absolutely can! Both methods work perfectly. FOIL method: (x-3)(x-3) = x² - 3x - 3x + 9 = x² - 6x + 9. The formula (a-b)² = a² - 2ab + b² is just faster once you memorize it.

What does 'standard form' actually mean?

Standard form for a quadratic function is $f(x) = ax^2 + bx + c$ where terms are arranged from highest to lowest degree. No parentheses, no factored terms - just expanded and simplified!

How do I remember the (a-b)² formula?

Think of it as 'square the first, minus twice the product, plus square the last'. Or remember: (a-b)² = a² - 2ab + b². Practice with simple numbers like (x-2)² first!

What if I get different coefficients in my answer?

Double-check your arithmetic! The most common errors are: forgetting the negative sign in -2ab, miscalculating 3² = 9, or missing the extra +x term. Work step-by-step and combine like terms carefully.

Can I leave my answer as (x-3)² + x?

No, that's vertex form, not standard form. The question specifically asks for standard form, which must be fully expanded as $ax^2 + bx + c$ .

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Transforming Quadratics: Find the Standard Form of f(x) = (x-3)² + x

Quadratic Expansion with Linear Terms

❤️ 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

Why can't I just multiply (x-3)(x-3) instead of using the formula?

What does 'standard form' actually mean?

How do I remember the (a-b)² formula?

What if I get different coefficients in my answer?

Can I leave my answer as (x-3)² + x?

🌟 Unlock Your Math Potential

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