Solve the Equation: x²+(x-2)²=2(x+1)² with Multiple Squared Terms

Q: Why does this quadratic equation become linear after expanding?

When you expand all terms, the $ x^2 $ terms cancel out! You get $ 2x^2 - 4x + 4 = 2x^2 + 4x + 2 $, and subtracting $ 2x^2 $ from both sides leaves only linear terms.

Q: How do I remember the perfect square formula?

Think of it as FOIL in reverse: $ (x-2)^2 $ means $ (x-2)(x-2) $. First terms: $ x^2 $, Outer + Inner: $ -2x - 2x = -4x $, Last terms: $ (-2)(-2) = 4 $.

Q: What if I expand incorrectly and get the wrong answer?

Always check your work! Substitute your answer back into the original equation. If $ x = \frac{1}{4} $ doesn't make both sides equal, you made an expansion error.

Q: Can I solve this without expanding everything?

You could try, but expanding first is usually easier. It converts the problem into a simpler linear equation once the $ x^2 $ terms cancel out.

Q: Why is the answer a fraction instead of a whole number?

Not all equations have integer solutions! Fractions are perfectly valid answers. Just make sure to reduce to lowest terms: $ \frac{2}{8} = \frac{1}{4} $.

Quadratic Equations with Algebraic Simplification

Solve the following problem:

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

❤️ 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 Find X

00:04 Use shortened multiplication formulas to expand all brackets

00:21 Calculate the multiplications and squares

00:38 Group like terms

00:58 Properly expand brackets, multiply by each term

01:06 Simplify where possible

01:11 Isolate X

01:32 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following problem:

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

Step-by-step solution

Proceed to solve the equation. First, we'll simplify the algebraic expressions by applying the perfect square binomial formula:

$(a\pm b)^2=a^2\pm2ab+b^2$

Apply the mentioned formula and expand the parentheses in the expressions in the equation:

$x^2+(x-2)^2=2(x+1)^2 \\ x^2+x^2-2\cdot x\cdot2+2^2=2(x^2+2\cdot x\cdot1+1^2)\\ x^2+x^2-4x+4=2(x^2+2x+1)\\ x^2+x^2-4x+4=2x^2+4x+2\\$ In the final stage, we used the distributive property on the right side of the equation.

Continue to combine like terms, by moving the terms between sides. We observe that the squared term cancels out, 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:

$x^2+x^2-4x+4=2x^2+4x+2 \\ -8x=-2\hspace{8pt}\text{/}:(-8)\\ \boxed{x=\frac{2}{8}=\frac{1}{4}}$

In the final stage, we reduced the fraction that was obtained as the solution for $x$ .

Therefore, the correct answer is answer A.

Final Answer

$x=\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Expansion: Use $(a±b)^2 = a^2 ± 2ab + b^2$ for all squared terms
Technique: Expand $(x-2)^2 = x^2 - 4x + 4$ and $2(x+1)^2 = 2x^2 + 4x + 2$
Check: Substitute $x = \frac{1}{4}$ : both sides equal $\frac{25}{8}$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute coefficients to expanded terms
Don't expand $2(x+1)^2$ as $x^2 + 2x + 1$ = wrong coefficients! You must multiply the 2 through every term after expanding. Always distribute coefficients: $2(x^2 + 2x + 1) = 2x^2 + 4x + 2$ .

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 this quadratic equation become linear after expanding?

When you expand all terms, the $x^2$ terms cancel out! You get $2x^2 - 4x + 4 = 2x^2 + 4x + 2$ , and subtracting $2x^2$ from both sides leaves only linear terms.

How do I remember the perfect square formula?

Think of it as FOIL in reverse: $(x-2)^2$ means $(x-2)(x-2)$ . First terms: $x^2$ , Outer + Inner: $-2x - 2x = -4x$ , Last terms: $(-2)(-2) = 4$ .

What if I expand incorrectly and get the wrong answer?

Always check your work! Substitute your answer back into the original equation. If $x = \frac{1}{4}$ doesn't make both sides equal, you made an expansion error.

Can I solve this without expanding everything?

You could try, but expanding first is usually easier. It converts the problem into a simpler linear equation once the $x^2$ terms cancel out.

Why is the answer a fraction instead of a whole number?

Not all equations have integer solutions! Fractions are perfectly valid answers. Just make sure to reduce to lowest terms: $\frac{2}{8} = \frac{1}{4}$ .

🌟 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