Solve the Equation: 2a(a-5) = (a+3)² + (a-3)²

Algebraic Expansion with Perfect Squares

Find a a given that

2a(a−5)=(a+3)2+(a−3)2 2a(a-5)=(a+3)^2+(a-3)^2

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find A
00:03 Open parentheses properly, multiply by each factor
00:17 Use shortened multiplication formulas to open parentheses
00:52 Calculate the squares and products
01:03 Reduce what we can
01:18 Collect like terms
01:28 Reduce what we can
01:33 Isolate A
01:40 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Find a a given that

2a(a−5)=(a+3)2+(a−3)2 2a(a-5)=(a+3)^2+(a-3)^2

2

Step-by-step solution

To solve this problem, we'll follow these steps:

  • Step 1: Expand both squared terms on the right side of the equation
  • Step 2: Simplify the terms and combine like terms
  • Step 3: Solve the simplified equation for a a

Let's now work through each step:

Step 1: Expand (a+3)2(a+3)^2 and (a−3)2(a-3)^2.
We know:
(a+3)2=a2+6a+9(a+3)^2 = a^2 + 6a + 9
(a−3)2=a2−6a+9(a-3)^2 = a^2 - 6a + 9

Step 2: Combine the expansions:
(a+3)2+(a−3)2=(a2+6a+9)+(a2−6a+9)=2a2+18(a+3)^2 + (a-3)^2 = (a^2 + 6a + 9) + (a^2 - 6a + 9) = 2a^2 + 18.

Step 3: Now, equate to the left side and simplify:
The left side of the equation is given as 2a(a−5)=2a2−10a2a(a-5) = 2a^2 - 10a.

Equating both sides:
2a2−10a=2a2+182a^2 - 10a = 2a^2 + 18

Subtract 2a22a^2 from both sides:
−10a=18-10a = 18

Divide by −10-10 to solve for aa:
a=18−10=−1.8a = \frac{18}{-10} = -1.8

Therefore, the solution to the problem is a=−1.8 a = -1.8 .

3

Final Answer

−1.8 -1.8

Key Points to Remember

Essential concepts to master this topic
  • Expansion: Use formulas like (a±b)2=a2±2ab+b2 (a±b)^2 = a^2 ± 2ab + b^2
  • Technique: (a+3)2+(a−3)2=2a2+18 (a+3)^2 + (a-3)^2 = 2a^2 + 18 after expanding and combining
  • Check: Substitute a=−1.8 a = -1.8 : both sides equal 24.48 24.48 ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to expand squared terms properly
    Don't just square the first and last terms like (a+3)2=a2+9 (a+3)^2 = a^2 + 9 = wrong answer! This misses the middle term 6a and leads to completely incorrect solutions. Always use the full formula (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 to expand squared binomials.

Practice Quiz

Test your knowledge with interactive questions

\( (4b-3)(4b-3) \)

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why do the middle terms cancel when I add (a+3)2 (a+3)^2 and (a−3)2 (a-3)^2 ?

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Great observation! When you expand (a+3)2=a2+6a+9 (a+3)^2 = a^2 + 6a + 9 and (a−3)2=a2−6a+9 (a-3)^2 = a^2 - 6a + 9 , the middle terms are +6a and -6a, which are opposites and cancel out completely!

How do I remember the perfect square formula?

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Think "First, twice, last": (a+b)2=a2+2ab+b2 (a+b)^2 = a^2 + 2ab + b^2 . The middle term is always twice the product of the two terms. Practice with simple examples like (x+2)2=x2+4x+4 (x+2)^2 = x^2 + 4x + 4 .

What if I get confused with all the algebra steps?

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Take it one step at a time! First expand everything, then collect like terms on each side, and finally solve the simplified equation. Write each step clearly to avoid mixing up terms.

Can I solve this problem without expanding the squares?

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While expanding is the standard method, you could also substitute specific values to check answers. However, algebraic expansion is the most reliable way to find the exact solution systematically.

Why does my calculator show -1.8 instead of a fraction?

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Both −1.8 -1.8 and −95 -\frac{9}{5} are the same value! Your calculator shows the decimal form, while the fraction is the exact form. Either representation is correct.

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