A Square Plus B Square Formula – Definition & Solved Examples

If you’ve ever looked at a math problem and wondered how to find the total of two squared numbers without knowing the numbers themselves, you’re looking for the a square plus b square formula. In algebra, this isn’t just a dry rule; it’s a clever shortcut. It allows you to solve for a sum of squares by using other pieces of information you might already have, such as the total sum of the two numbers or the result of multiplying them together.

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What is the a^2 + b^2 Formula?

The term a square plus b square refers to an algebraic identity. While many students quickly learn how to handle the "difference of squares," the "sum of squares" is a little different because it doesn't have a direct, single-bracket expansion like (a+b)(a-b). Instead, the a^2 + b^2 formula is derived by rearranging the parts of a squared binomial.

Depending on what "clues" your math problem gives you, there are two main ways to write this formula:

• When you know the sum: a^2 + b^2 = (a + b)^2 - 2ab

• When you know the difference: a^2 + b^2 = (a - b)^2 + 2ab

The beauty of these two options is flexibility. If your problem provides the sum (a + b), you use the first version. If it provides the difference (a - b), you switch to the second. Both paths lead to the exact same answer.

How to Work Out the a square plus b square Identity

Formulae are much easier to remember when you see the "why" behind them. Let’s look at what is the a^2 + b^2 formula by pulling apart standard algebraic squares.

Deriving the formula from (a + b)^2

1. Think about the standard expansion: (a + b)^2 = a^2 + 2ab + b^2.

2. Our goal is to get a^2 and b^2 alone on one side.

3. To do that, we simply move the "2ab" to the other side by subtracting it.

4. This gives us our first tool: a^2 + b^2 = (a + b)^2 - 2ab.

Deriving the formula from (a - b)^2

1. Now, look at the expansion for a difference: (a - b)^2 = a^2 - 2ab + b^2.

2. To isolate our squares this time, we need to cancel out the "-2ab".

3. We do this by adding 2ab to the other side.

4. This gives us our second tool: a^2 + b^2 = (a - b)^2 + 2ab.

Where do we use this in real life?

This formula is far more than just "school work." It is a heavy lifter in several practical fields:

• Engineering and Architecture: It’s the engine inside the Pythagorean theorem (a^2 + b^2 = c^2), which builders use to make sure walls are perfectly straight and roofs are angled correctly.

• Computer Science: Programmers use these identities to calculate distances between objects on a screen or to handle complex data structures.

• Physics: It helps calculate the resulting force when two different powers are pushing on an object at the same time.

Read More - (a + b)³ Formula: A Plus B Whole Cube Formula

A square plus b square example: Let’s Solve Some Problems

The best way to get this into your brain is to see it in action. Here are three common ways you might use the a square plus b square formula.

Example 1: Using the Sum and the Product

The Question: If a + b = 10 and ab = 21, what is the value of a^2 + b^2?

• The Approach: Since we have the sum (10), we use: a^2 + b^2 = (a + b)^2 - 2ab.

• The Math: 1. Square the sum: (10)^2 = 100.
  2. Double the product: 2 * 21 = 42.
  3. Subtract: 100 - 42 = 58.

• The Answer: 58.

Example 2: Using the Difference and the Product

The Question: If a - b = 5 and ab = 6, find the value of a^2 + b^2?

• The Approach: Since we have the difference (5), we use: a^2 + b^2 = (a - b)^2 + 2ab.

• The Math:

1. Square the difference: (5)^2 = 25.

2. Double the product: 2 * 6 = 12.

3. Add them together: 25 + 12 = 37.

• The Answer: 37.

Read More - A^2-B^2 Formula: Proof with Examples

Example 3: The "Fraction Trick"

The Question: Find x^2 + 1/x^2 if you are told that x + 1/x = 4.

• The Approach: This looks scary, but it’s just the same formula. Here, "a" is x and "b" is 1/x. If you multiply them (ab), they cancel out to equal 1!

• The Math:

1. Use the sum version: (x + 1/x)^2 - 2(x * 1/x).

2. Substitute the values: (4)^2 - 2(1).

3. Calculate: 16 - 2 = 14.

• The Answer: 14.

A Simple Memory Trick

If you find yourself mixing up the plus and minus signs, remember the "Opposites Rule":

• If you start with a plus in the brackets, you subtract at the end.

• If you start with a minus in the brackets, you add at the end.
  Think of it as keeping the equation in balance!

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