Solve for x in (x+1)(2x+1) = 2x²+4: Expanding Binomial Products

Q: Why did the x² terms disappear?

Great observation! The $ x^2 $ terms canceled out because we had $ 2x^2 $ on both sides. When we subtract $ 2x^2 $ from both sides, they disappear, leaving us with a simpler linear equation!

Q: What if I make a mistake when expanding (x+1)(2x+1)?

Use FOIL carefully: First terms (x·2x), Outer terms (x·1), Inner terms (1·2x), Last terms (1·1). This gives you $ 2x^2+x+2x+1 $. Double-check by adding: 2x²+3x+1.

Q: Can I solve this without expanding the left side?

It's much harder that way! Expanding reveals that the $ x^2 $ terms cancel, making this a simple linear equation. Without expanding, you'd miss this crucial simplification.

Q: What if I got x = 3 instead of x = 1?

Check your algebra! A common mistake is writing $ x + 2x = 3x $ as $ 2x $ instead of $ 3x $. Always carefully combine like terms and verify by substituting your answer back into the original equation.

Binomial Expansion with Quadratic Simplification

Solve for x:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find X.

00:13 First, open the parentheses carefully. Multiply each part inside by every other part.

00:28 Next, simplify what you can. Gather all the like terms.

00:38 Now, let's isolate X. Move everything else to the other side.

00:51 And that's how we find the solution! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve for x:

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

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 will therefore apply the mentioned law and open the parentheses in the expression in the equation:

$(x+1)(2x+1)=2x^2+4 \\ 2x^2+x+2x+1=2x^2+4 \\$ We'll continue and combine like terms, by moving terms, then - we can notice that the term with the squared power 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:

$2x^2+x+2x+1=2x^2+4\\ 3x=3\hspace{8pt}\text{/}:3\\ \boxed{x=1}$ Therefore, the correct answer is answer B.

Final Answer

$x=1$

Key Points to Remember

Essential concepts to master this topic

Distribution: Apply FOIL method: (a+b)(c+d) = ac+ad+bc+bd
Technique: Combine like terms: $2x^2+x+2x+1 = 2x^2+3x+1$
Check: Substitute x=1: (1+1)(2·1+1) = 2·1²+4 gives 6=6 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand both sides completely
Don't expand only the left side and leave the right side as is = missing the cancellation! This prevents you from seeing that the quadratic terms cancel out. Always expand both sides completely, then combine like terms to reveal the true nature of the equation.

Practice Quiz

Test your knowledge with interactive questions

$$ (x+y)(x-y)= $$

FAQ

Everything you need to know about this question

Why did the x² terms disappear?

Great observation! The $x^2$ terms canceled out because we had $2x^2$ on both sides. When we subtract $2x^2$ from both sides, they disappear, leaving us with a simpler linear equation!

What if I make a mistake when expanding (x+1)(2x+1)?

Use FOIL carefully: First terms (x·2x), Outer terms (x·1), Inner terms (1·2x), Last terms (1·1). This gives you $2x^2+x+2x+1$ . Double-check by adding: 2x²+3x+1.

How do I know this became a linear equation?

After expanding and simplifying, the highest power of x that remains determines the equation type. Since our $x^2$ terms canceled out, we're left with $3x = 3$ , which is linear (first degree).

Can I solve this without expanding the left side?

It's much harder that way! Expanding reveals that the $x^2$ terms cancel, making this a simple linear equation. Without expanding, you'd miss this crucial simplification.

What if I got x = 3 instead of x = 1?

Check your algebra! A common mistake is writing $x + 2x = 3x$ as $2x$ instead of $3x$ . Always carefully combine like terms and verify by substituting your answer back into the original equation.

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