Understanding the Algebra Problem

The problem asks us to find the value of the product ab given two pieces of information about variables a and b:

• The sum of a and b is 9: \(\text{a} + \text{b} = 9\)
• The sum of the squares of a and b is 53: \(\text{a}^2 + \text{b}^2 = 53\)

We need to use these given equations to determine the specific value of ab.

Using Algebraic Identities to Solve for ab

This type of problem can be solved using a fundamental algebraic identity that relates the sum of two terms, the sum of their squares, and their product. The identity is:

\((\text{a} + \text{b})^2 = \text{a}^2 + \text{b}^2 + 2\text{ab}\)

This identity shows that the square of the sum of a and b is equal to the sum of their squares plus twice their product (2ab). We can rearrange this identity to solve for 2ab:

\(2\text{ab} = (\text{a} + \text{b})^2 - (\text{a}^2 + \text{b}^2)\)

Now, we can substitute the values given in the problem into this rearranged identity.

Step-by-Step Calculation of ab

We are given:

• \(\text{a} + \text{b} = 9\)
• \(\text{a}^2 + \text{b}^2 = 53\)

Substitute these values into the rearranged identity \(2\text{ab} = (\text{a} + \text{b})^2 - (\text{a}^2 + \text{b}^2)\):

\(2\text{ab} = (9)^2 - (53)\)

Calculate the square of 9:

\(9^2 = 9 \times 9 = 81\)

Substitute this value back into the equation:

\(2\text{ab} = 81 - 53\)

Perform the subtraction:

\(81 - 53 = 28\)

So, we have:

\(2\text{ab} = 28\)

To find the value of ab, divide both sides of the equation by 2:

\(\text{ab} = \frac{28}{2}\)

\(\text{ab} = 14\)

Thus, the value of ab is 14.

Verification

Let's check if our value of ab = 14 is consistent with the original equations using the identity \((a+b)^2 = a^2 + b^2 + 2ab\):

Substitute the given values and our calculated value for 2ab:

\((\text{a} + \text{b})^2 = 9^2 = 81\)

\(\text{a}^2 + \text{b}^2 + 2\text{ab} = 53 + 2(14) = 53 + 28 = 81\)

Since \(81 = 81\), our calculated value for ab is correct.

Finding the Value of ab Summary

Here is a summary of the steps taken to find the value of ab:

1. Identify the given equations: \(\text{a} + \text{b} = 9\) and \(\text{a}^2 + \text{b}^2 = 53\).
2. Recall or use the algebraic identity: \((\text{a} + \text{b})^2 = \text{a}^2 + \text{b}^2 + 2\text{ab}\).
3. Rearrange the identity to solve for \(2\text{ab}\): \(2\text{ab} = (\text{a} + \text{b})^2 - (\text{a}^2 + \text{b}^2)\).
4. Substitute the given values for \((\text{a} + \text{b})\) and \((\text{a}^2 + \text{b}^2)\): \(2\text{ab} = (9)^2 - (53)\).
5. Calculate \((9)^2\): \(81\).
6. Substitute the squared value: \(2\text{ab} = 81 - 53\).
7. Perform the subtraction: \(2\text{ab} = 28\).
8. Divide by 2 to find ab: \(\text{ab} = \frac{28}{2} = 14\).

The value of ab is 14.

Revision Table: Key Algebraic Identities

Additional Information: Applications of Algebraic Identities

Algebraic identities are powerful tools used extensively in mathematics and other fields. They help simplify expressions, solve equations, and factor polynomials. For example:

• Simplifying Calculations: Identities like \((a+b)^2\) or \((a-b)^2\) can make squaring numbers easier, e.g., \(103^2 = (100+3)^2 = 100^2 + 2(100)(3) + 3^2 = 10000 + 600 + 9 = 10609\).
• Solving Quadratic Equations: Completing the square, a method to solve quadratic equations, is based on the square of a sum or difference identity.
• Factorization: The difference of squares identity (\(\text{x}^2 - \text{y}^2 = (\text{x} - \text{y})(\text{x} + \text{y})\)) is crucial for factoring many algebraic expressions.
• Calculus: Identities are used to simplify functions before differentiation or integration.
• Physics and Engineering: Many formulas and equations in physics and engineering rely on algebraic manipulations simplified by identities.

Understanding these identities is fundamental for success in algebra and related mathematical areas.