Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product

Q: Why can't I just take the square root of both sides?

You can't take the square root directly because the right side isn't a perfect square. $ (x+2)(x-1) $ doesn't simplify to something squared, so you must expand both sides first.

Q: How do I remember the perfect square formula?

Think "First squared, minus twice the product, plus last squared": $ (a-b)^2 = a^2 - 2ab + b^2 $. For $ (x-4)^2 $: $ x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16 $.

Q: What if the x² terms don't cancel out?

If $ x^2 $ terms don't cancel, you'll have a quadratic equation instead of linear. You'd need to use factoring, completing the square, or the quadratic formula to solve it.

Q: How do I expand (x+2)(x-1) without making mistakes?

Use FOIL: First + Outer + Inner + Last. $ (x+2)(x-1) = x \cdot x + x \cdot (-1) + 2 \cdot x + 2 \cdot (-1) = x^2 - x + 2x - 2 = x^2 + x - 2 $.

Q: Why do we get a linear equation from two quadratic expressions?

After expanding both sides, the $ x^2 $ terms are identical ($ x^2 $ on both sides), so they cancel out when we subtract. This leaves us with a simpler linear equation to solve!

Quadratic Equations with Binomial Expansion

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 We'll use shortened multiplication formulas to open the brackets

00:08 Open brackets properly, multiply each factor by each term

00:33 Simplify what we can

00:42 Isolate X

00:57 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

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

Step-by-step solution

To solve the equation $(x-4)^2 = (x+2)(x-1)$ , follow these detailed steps:

Step 1: Expand the left side of the equation using the square of a binomial formula: $(x-4)^2 = x^2 - 8x + 16$ .
Step 2: Expand the right side using the distributive property: $(x+2)(x-1) = x(x-1) + 2(x-1) = x^2 - x + 2x - 2 = x^2 + x - 2$ .
Step 3: Set the expanded forms equal to each other: $x^2 - 8x + 16 = x^2 + x - 2$ .
Step 4: Subtract $x^2$ from both sides to simplify: $-8x + 16 = x - 2$ .
Step 5: Move all terms involving $x$ to one side and constant terms to the other: $-8x - x = -2 - 16$ .
Step 6: Combine like terms: $-9x = -18$ .
Step 7: Solve for $x$ by dividing both sides by $-9$ : $x = 2$ .

Therefore, the solution to the problem is $x = 2$ .

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Expansion: Use $(a-b)^2 = a^2 - 2ab + b^2$ for perfect squares
Technique: $(x-4)^2 = x^2 - 8x + 16$ and $(x+2)(x-1) = x^2 + x - 2$
Check: Substitute x = 2: $(2-4)^2 = 4$ and $(2+2)(2-1) = 4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the middle term when expanding perfect squares
Don't expand $(x-4)^2$ as just $x^2 + 16$ = missing the -8x term! This loses crucial information and leads to completely wrong answers. Always include the middle term: $(x-4)^2 = x^2 - 8x + 16$ .

Practice Quiz

Test your knowledge with interactive questions

It is possible to use the distributive property to simplify the expression below?

What is its simplified form?

$$ (ab)(c d) $$

$$

FAQ

Everything you need to know about this question

Why can't I just take the square root of both sides?

You can't take the square root directly because the right side isn't a perfect square. $(x+2)(x-1)$ doesn't simplify to something squared, so you must expand both sides first.

How do I remember the perfect square formula?

Think "First squared, minus twice the product, plus last squared": $(a-b)^2 = a^2 - 2ab + b^2$ . For $(x-4)^2$ : $x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16$ .

What if the x² terms don't cancel out?

If $x^2$ terms don't cancel, you'll have a quadratic equation instead of linear. You'd need to use factoring, completing the square, or the quadratic formula to solve it.

How do I expand (x+2)(x-1) without making mistakes?

Use FOIL: First + Outer + Inner + Last. $(x+2)(x-1) = x \cdot x + x \cdot (-1) + 2 \cdot x + 2 \cdot (-1) = x^2 - x + 2x - 2 = x^2 + x - 2$ .

Why do we get a linear equation from two quadratic expressions?

After expanding both sides, the $x^2$ terms are identical ( $x^2$ on both sides), so they cancel out when we subtract. This leaves us with a simpler linear equation to solve!

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Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product

Quadratic Equations with Binomial Expansion

❤️ 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 take the square root of both sides?

How do I remember the perfect square formula?

What if the x² terms don't cancel out?

How do I expand (x+2)(x-1) without making mistakes?

Why do we get a linear equation from two quadratic expressions?

🌟 Unlock Your Math Potential

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