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

Q: Why does the x² term cancel out on both sides?

When you expand both sides, you get $ x^2 - 8x + 16 = x^2 + x - 2 $. Since both sides have $ x^2 $, they cancel out when you subtract, leaving a linear equation!

Q: How do I remember the squared binomial formula?

Think "First squared, minus twice the product, plus last squared": $ (a-b)^2 = a^2 - 2ab + b^2 $. Practice with simple examples like $ (x-3)^2 $.

Q: What's the FOIL method for (x+2)(x-1)?

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

Q: Can I solve this without expanding everything?

While expanding is the standard method, you could try substituting answer choices, but expanding helps you understand the algebra and works for any similar problem.

Q: Why is the final answer x = 2 and not one of the other choices?

After simplifying to $ -9x = -18 $, divide both sides by -9 to get $ x = 2 $. Always verify by substituting back into the original equation!

Quadratic Equations with Squared Binomials

Solve for x:

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's find X.

00:12 We'll use the multiplication formulas to expand the parentheses.

00:16 Carefully open the parentheses. Multiply each term with every other term.

00:32 Now let's solve the multiplication and square any terms where needed.

00:40 Simplify by combining like terms.

00:46 Next, let's isolate the variable X.

01:10 And there you have it, that's the solution to our problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

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

Step-by-step solution

Let's solve the equation, first we'll simplify the algebraic expressions using the extended distribution law:

$(a+b)(c+d)=ac+ad+bc+bd$ We'll use the shortened multiplication formula for a squared binomial:

$(a\pm b)^2=a^2\pm2ab+b^2$ We'll therefore apply the law and formula mentioned and open the parentheses in the expressions in the equation:

$(x-4)^2=(x+2)(x-1) \\ x^2-2\cdot x\cdot4+4^2=x^2-x+2x-2 \\ x^2-8x+16=x^2+x-2$ We'll continue and combine like terms, by moving terms between sides - we can notice that the squared term cancels out and therefore it's a first-degree equation, which we'll solve by isolating the variable term on one side and dividing both sides of the equation by its coefficient:

$x^2-8x+16=x^2+x-2\\ -9x=-18\hspace{8pt}\text{/}:(-9)\\ \boxed{x=2}$ Therefore the correct answer is answer A.

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Expansion: Use $(a-b)^2 = a^2 - 2ab + b^2$ and FOIL method
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

Incorrectly expanding the squared binomial
Don't expand $(x-4)^2$ as $x^2 - 16$ = missing the middle term! This forgets the -2ab term in the formula. Always use $(a-b)^2 = a^2 - 2ab + b^2$ to get $x^2 - 8x + 16$ .

Practice Quiz

Test your knowledge with interactive questions

Declares the given expression as a sum

$$ (7b-3x)^2 $$

FAQ

Everything you need to know about this question

Why does the x² term cancel out on both sides?

When you expand both sides, you get $x^2 - 8x + 16 = x^2 + x - 2$ . Since both sides have $x^2$ , they cancel out when you subtract, leaving a linear equation!

How do I remember the squared binomial formula?

Think "First squared, minus twice the product, plus last squared": $(a-b)^2 = a^2 - 2ab + b^2$ . Practice with simple examples like $(x-3)^2$ .

What's the FOIL method for (x+2)(x-1)?

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

Can I solve this without expanding everything?

While expanding is the standard method, you could try substituting answer choices, but expanding helps you understand the algebra and works for any similar problem.

Why is the final answer x = 2 and not one of the other choices?

After simplifying to $-9x = -18$ , divide both sides by -9 to get $x = 2$ . Always verify by substituting back into the original equation!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Short Multiplication Formulas questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime