Math Formula a2+b2

Understanding the Formula \(a^2 + b^2\)

In mathematics, the expression \(a^2 + b^2\) represents the sum of the squares of two numbers, \(a\) and \(b\). This formula is fundamental in various branches of mathematics, including algebra, geometry, and trigonometry. It is commonly encountered in problems involving the Pythagorean theorem, quadratic equations, and coordinate geometry.

Breaking Down the Formula

The formula \(a^2 + b^2\) is composed of two terms:

• \(a^2\): This represents the square of the number \(a\). Squaring a number means multiplying the number by itself. Thus, \(a^2 = a \times a\).
• \(b^2\): This represents the square of the number \(b\). Similarly, \(b^2 = b \times b\).

Applications of \(a^2 + b^2\)

1. Pythagorean Theorem

The Pythagorean theorem is one of the most famous applications of the formula \(a^2 + b^2\). It states that in a right-angled triangle, the square of the hypotenuse (\(c\)) is equal to the sum of the squares of the other two sides (\(a\) and \(b\)):

\[ c^2 = a^2 + b^2 \]

Example:

Consider a right-angled triangle with sides of length 3 and 4. To find the length of the hypotenuse:

\[ c^2 = 3^2 + 4^2 \]

\[ c^2 = 9 + 16 \]

\[ c^2 = 25 \]

\[ c = \sqrt{25} \]

\[ c = 5 \]

Thus, the hypotenuse is 5 units long.

2. Distance Formula

In coordinate geometry, the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) in a plane can be calculated using the formula derived from \(a^2 + b^2\):

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Example:

Find the distance between points (1, 2) and (4, 6):

\[ d = \sqrt{(4 - 1)^2 + (6 - 2)^2} \]

\[ d = \sqrt{3^2 + 4^2} \]

\[ d = \sqrt{9 + 16} \]

\[ d = \sqrt{25} \]

\[ d = 5 \]

The distance between the points is 5 units.

3. Quadratic Equations

In algebra, the expression \(a^2 + b^2\) can appear in quadratic equations. Although it does not directly form a standard quadratic equation, it can be part of expressions that need simplification or factorization.

Example:

Simplify \( (a + b)^2 - 2ab \):

Using the binomial expansion:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

So,

\[ (a + b)^2 - 2ab = a^2 + 2ab + b^2 - 2ab \]

\[ = a^2 + b^2 \]

Thus, \( (a + b)^2 - 2ab = a^2 + b^2 \).

Visual Representation

To visualize \(a^2 + b^2\), consider the following geometric interpretation:

• Imagine two squares, one with side length \(a\) and the other with side length \(b\).
• The area of the first square is \(a^2\) and the area of the second square is \(b^2\).

When you sum these areas, you get \(a^2 + b^2\), representing the total area of the two squares combined.

Conclusion

The formula \(a^2 + b^2\) is a versatile and widely used expression in mathematics. Its applications range from the Pythagorean theorem in geometry to distance calculations in coordinate geometry and simplifications in algebra. Understanding and mastering this formula is essential for solving various mathematical problems effectively.

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