Convert (x-5)² - 10 to its Standard Quadratic Form

Q: Why can't I just distribute the square to get x² - 25?

Because squaring isn't the same as distributing! When you square $(x-5)$, you're multiplying $(x-5) \times (x-5)$, which gives you three terms, not two.

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

Vertex form is $f(x) = a(x-h)^2 + k$ and shows the vertex clearly. Standard form is $f(x) = ax^2 + bx + c$ and makes it easy to see the y-intercept and use the quadratic formula.

Q: How do I remember the perfect square formula?

Think "First, Opposite, Last": $(x-5)^2$ gives you x² (first squared), -10x (opposite of middle term doubled), +25 (last squared). The middle term is always twice the product!

Q: Can I use FOIL instead of the formula?

Absolutely! FOIL works great: $(x-5)(x-5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25$. Use whichever method feels more comfortable to you.

Q: What if I get the wrong sign in my final answer?

Double-check your constant term combination! In this problem: +25 from the expansion, then -10 from the original equation gives +15. Sign errors usually happen when combining these final terms.

Quadratic Functions with Vertex to Standard Form

Find the standard representation of the following function

$f(x)=(x-5)^2-10$

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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

00:03 Open parentheses according to the shortened multiplication formulas

00:11 Calculate powers and multiplications

00:29 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

$f(x)=(x-5)^2-10$

Step-by-step solution

To convert the quadratic function from vertex form to standard form, execute the following steps:

Step 1: Begin with the given vertex form $f(x) = (x-5)^2 - 10$ .
Step 2: Expand $(x-5)^2$ using the formula $(a-b)^2 = a^2 - 2ab + b^2$ , which results in:

$(x-5)^2 = x^2 - 2 \cdot x \cdot 5 + 5^2 = x^2 - 10x + 25$ .

Step 3: Replace the expanded form into the original function:

$f(x) = x^2 - 10x + 25 - 10$ .

Step 4: Combine like terms:

$f(x) = x^2 - 10x + 15$ .

Therefore, the standard form of the function is $f(x) = x^2 - 10x + 15$ .

Comparing with the given choices, the correct option is:

$f(x) = x^2 - 10x + 15$

Final Answer

$f(x)=x^2-10x+15$

Key Points to Remember

Essential concepts to master this topic

Rule: Expand perfect square trinomial using $(a-b)^2 = a^2 - 2ab + b^2$
Technique: $(x-5)^2 = x^2 - 10x + 25$ , then combine constants
Check: Substitute test value like x=0: both forms give same result ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term when expanding
Don't expand (x-5)² as just x² + 25 = missing the -10x term! This creates a completely wrong equation. Always use the full formula: (a-b)² = a² - 2ab + b² to get all three terms.

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 get x² - 25?

Because squaring isn't the same as distributing! When you square $(x-5)$ , you're multiplying $(x-5) \times (x-5)$ , which gives you three terms, not two.

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

Vertex form is $f(x) = a(x-h)^2 + k$ and shows the vertex clearly. Standard form is $f(x) = ax^2 + bx + c$ and makes it easy to see the y-intercept and use the quadratic formula.

How do I remember the perfect square formula?

Think "First, Opposite, Last": $(x-5)^2$ gives you x² (first squared), -10x (opposite of middle term doubled), +25 (last squared). The middle term is always twice the product!

Can I use FOIL instead of the formula?

Absolutely! FOIL works great: $(x-5)(x-5) = x^2 - 5x - 5x + 25 = x^2 - 10x + 25$ . Use whichever method feels more comfortable to you.

What if I get the wrong sign in my final answer?

Double-check your constant term combination! In this problem: +25 from the expansion, then -10 from the original equation gives +15. Sign errors usually happen when combining these final terms.

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Convert (x-5)² - 10 to its Standard Quadratic Form

Quadratic Functions with Vertex to Standard Form

❤️ 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 distribute the square to get x² - 25?

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

How do I remember the perfect square formula?

Can I use FOIL instead of the formula?

What if I get the wrong sign in my final answer?

🌟 Unlock Your Math Potential

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