Transform y=3x+10 into (x-4)²: Finding the Missing Expression

Q: Why do I need to expand (x-4)² first?

You need to see what the target expression looks like in standard form! Expanding $ (x-4)^2 $ gives you $ x^2 - 8x + 16 $, which you can then compare with $ 3x + 10 $.

Q: How do I know what to subtract from what?

Think of it as: What do I add to get from A to B? The answer is B - A. So subtract your starting expression from your target expression.

Q: What if I get a negative coefficient?

That's completely normal! In this problem, we get $ -11x $ because we're subtracting $ 3x $ from $ -8x $. Negative coefficients are just part of algebra.

Q: Can I check my answer?

Absolutely! Add your answer to the original expression: $ (3x + 10) + (x^2 - 11x + 6) $. If you get $ x^2 - 8x + 16 = (x-4)^2 $, you're correct!

Q: Why can't I just work backwards from the answers?

While that might work for multiple choice, understanding the method is crucial! This transformation technique appears in many algebra problems, so learning the proper steps helps you solve similar problems independently.

Algebraic Transformation with Expression Addition

$y=3x+10$

Which expression should be added to Y so that :

$y=(x-4)^2$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the correct expression so that the equation is satisfied

00:05 Use the short multiplication formulas to expand the brackets

00:16 Find the difference between the expressions

00:29 Add the differences to the expression

00:32 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

$y=3x+10$

Which expression should be added to Y so that :

$y=(x-4)^2$

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Expand the expression $(x - 4)^2$ .
Step 2: Compare the expanded quadratic expression with $3x + 10$ .
Step 3: Determine the additional expression needed.

Let's carry out these steps:

Step 1: First, expand $(x - 4)^2$ . Using the formula for the square of a binomial, we have:

$(x - 4)^2 = x^2 - 2 \cdot x \cdot 4 + 4^2$

This simplifies to:

$x^2 - 8x + 16$

Step 2: We need to convert $y = 3x + 10$ into this expanded form. That means $3x + 10$ should become $x^2 - 8x + 16$ .

Step 3: The expression to add is the difference between $x^2 - 8x + 16$ and $3x + 10$ :

Subtract $3x + 10$ from $x^2 - 8x + 16$ :

$(x^2 - 8x + 16) - (3x + 10)$

This simplifies to:

$x^2 - 8x + 16 - 3x - 10 = x^2 - 11x + 6$

Therefore, the expression that should be added to $y$ is $x^2 - 11x + 6$ .

Final Answer

$x^2-11x+6$

Key Points to Remember

Essential concepts to master this topic

Expansion: Expand $(x-4)^2 = x^2 - 8x + 16$
Technique: Find difference: $(x^2 - 8x + 16) - (3x + 10) = x^2 - 11x + 6$
Check: Verify: $(3x + 10) + (x^2 - 11x + 6) = x^2 - 8x + 16 = (x-4)^2$ ✓

Common Mistakes

Avoid these frequent errors

Adding instead of finding the difference
Don't add the two expressions together = wrong transformation! This gives you a completely different quadratic. Always subtract the original expression from the target expression to find what needs to be added.

Practice Quiz

Test your knowledge with interactive questions

$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why do I need to expand (x-4)² first?

You need to see what the target expression looks like in standard form! Expanding $(x-4)^2$ gives you $x^2 - 8x + 16$ , which you can then compare with $3x + 10$ .

How do I know what to subtract from what?

Think of it as: What do I add to get from A to B? The answer is B - A. So subtract your starting expression from your target expression.

What if I get a negative coefficient?

That's completely normal! In this problem, we get $-11x$ because we're subtracting $3x$ from $-8x$ . Negative coefficients are just part of algebra.

Can I check my answer?

Absolutely! Add your answer to the original expression: $(3x + 10) + (x^2 - 11x + 6)$ . If you get $x^2 - 8x + 16 = (x-4)^2$ , you're correct!

Why can't I just work backwards from the answers?

While that might work for multiple choice, understanding the method is crucial! This transformation technique appears in many algebra problems, so learning the proper steps helps you solve similar problems independently.

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