Simplify (x + √x)²: Expanding a Binomial with Square Root

Q: Why can't I just square each term separately?

Squaring binomials requires the full expansion formula! You can't just do x² + (√x)² because you're missing the crucial middle term $ 2ab $. Always use (a + b)² = a² + 2ab + b².

Q: How do I handle the square root in the middle term?

When calculating $ 2ab = 2 \times x \times \sqrt{x} $, remember that x × √x = x√x. So the middle term becomes $ 2x\sqrt{x} $, not 2x + 2√x!

Q: What does (√x)² equal?

The square and square root cancel each other out! So $ (\sqrt{x})^2 = x $. This is because √x means "what number times itself gives x" - so (√x)² = x.

Q: Why do we factor out x at the end?

Factoring helps simplify the expression and makes it match the answer choices! Since every term contains x as a factor: $ x^2 + 2x\sqrt{x} + x = x(x + 2\sqrt{x} + 1) $.

Q: Can I check my answer by expanding it back?

Yes! Take your final answer $ x(x + 2\sqrt{x} + 1) $ and distribute the x: you should get back to $ x^2 + 2x\sqrt{x} + x $. This confirms your expansion was correct!

Binomial Expansion with Square Root Terms

Simply the following expression:

$(x+\sqrt{x})^2$

❤️ 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

00:00 Simply

00:03 We'll use the abbreviated multiplication formulas to open the parentheses

00:11 We'll use this formula in our exercise where X equals A

00:20 and the root of X is B

00:24 We'll substitute according to the formula

00:42 A square root squared equals the number itself

00:50 We'll factor X squared into factors X and X

00:56 We'll find a common factor and take it out of the parentheses

01:04 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

Simply the following expression:

$(x+\sqrt{x})^2$

Step-by-step solution

To solve the problem, we undertake the following steps:

Identify the components $a = x$ and $b = \sqrt{x}$ in the expression $(x + \sqrt{x})^2$ .
Apply the square of sum formula: $(a + b)^2 = a^2 + 2ab + b^2$ .
Substitute the values:
- $a^2 = x^2$
- $2ab = 2 \times x \times \sqrt{x} = 2x\sqrt{x}$
- $b^2 = (\sqrt{x})^2 = x$
Simplifying gives: $x^2 + 2x\sqrt{x} + x$
Recognize that this can be factored to yield: $x(x + 2\sqrt{x} + 1)$

Thus, the simplified expression is $\boxed{x[x + 2\sqrt{x} + 1]}$ .

Final Answer

$x\lbrack x+2\sqrt{x}+1\rbrack$

Key Points to Remember

Essential concepts to master this topic

Formula: Use (a + b)² = a² + 2ab + b² pattern
Technique: Calculate 2x√x = 2x√x, then (√x)² = x
Check: Factor out x: x² + 2x√x + x = x(x + 2√x + 1) ✓

Common Mistakes

Avoid these frequent errors

Treating square root incorrectly in middle term
Don't calculate 2ab as 2x + 2√x = wrong expansion! This ignores the product rule for the middle term. Always multiply the terms together: 2 × x × √x = 2x√x.

Practice Quiz

Test your knowledge with interactive questions

Break down the expression into basic terms:

$$ 4x^2 + 6x $$

FAQ

Everything you need to know about this question

Why can't I just square each term separately?

Squaring binomials requires the full expansion formula! You can't just do x² + (√x)² because you're missing the crucial middle term $2ab$ . Always use (a + b)² = a² + 2ab + b².

How do I handle the square root in the middle term?

When calculating $2ab = 2 \times x \times \sqrt{x}$ , remember that x × √x = x√x. So the middle term becomes $2x\sqrt{x}$ , not 2x + 2√x!

What does (√x)² equal?

The square and square root cancel each other out! So $(\sqrt{x})^2 = x$ . This is because √x means "what number times itself gives x" - so (√x)² = x.

Why do we factor out x at the end?

Factoring helps simplify the expression and makes it match the answer choices! Since every term contains x as a factor: $x^2 + 2x\sqrt{x} + x = x(x + 2\sqrt{x} + 1)$ .

Can I check my answer by expanding it back?

Yes! Take your final answer $x(x + 2\sqrt{x} + 1)$ and distribute the x: you should get back to $x^2 + 2x\sqrt{x} + x$ . This confirms your expansion was correct!

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Algebraic Technique 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

Simplify (x + √x)²: Expanding a Binomial with Square Root

Binomial Expansion with Square Root Terms

❤️ 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 square each term separately?

How do I handle the square root in the middle term?

What does (√x)² equal?

Why do we factor out x at the end?

Can I check my answer by expanding it back?

🌟 Unlock Your Math Potential

More Questions

Factorization - Common Factor

Solve 3m(? + ?) = 6mn + 9m: Finding Missing Values in Algebraic Distribution

Solving for Variables: Calculate A and B in the Transformed Equation

Solve for Missing Values in 12ab(? + ?) = 24abc + 36

Solve for the Missing Factor in ?(5b-3) = 20b-15: Algebraic Equation Practice

Square of sum

Solve the Radical Equation: Find X in x + √x = -√x

Solve for X: Area vs. Perimeter in Two Altered Square Equations

Solve the System of Equations: Square Roots and xy=9

Simplify the Radical Fraction: √(2x²+4xy+2y²+(x+y)²)/(x+y)