Solving the Problem a2 b2 25, ab 12: A Comprehensive Guide

When faced with the problem of finding the values of a and b given the equations a2 b2 25 and ab 12, a step-by-step solution using algebraic identities and the properties of quadratic equations can be applied. This guide will walk you through the process, ensuring a clear and thorough understanding.

Step-by-Step Solution

Step 1: Utilize the Identity for Sum of Squares

One of the essential algebraic identities we can use is the relationship between the sum of squares and the product of two numbers, which states that:

a2 b2 (a b)2 - 2ab

Step 2: Substitute Known Values

Given the equations:

a2 b2 25 ab 12

We can substitute these into the identity:

a2 b2 (a b)2 - 2ab

Substituting the known values:

25 (a b)2 - 212

Simplify the equation:

25 (a b)2 - 24

(a b)2 49

Take the square root of both sides:

a b 7 or a b -7.

Step 3: Solve the System of Equations

With the values a b 7 and a b -7, we can solve for a and b using the product equation.

ab 12.

Case 1: a b 7

Let b 7 - a and substitute in the product equation:

a(7 - a) 12

7a - a2 12

a2 - 7a 12 0

This is a quadratic equation which can be factored as:

(a - 3)(a - 4) 0

Thus, the solutions are:

a 3 a 4

Substitute Back to Find b

For a 3, we get:

b 7 - 3 4

For a 4, we get:

b 7 - 4 3

Case 2: a b -7

Following a similar process, we find that the solutions for this case are:

a -3 a -4

Substitute Back to Find b

For a -3, we get:

b -7 - (-3) -4

For a -4, we get:

b -7 - (-4) -3

Final Result

The values of a and b that satisfy the given equations are:

a 3, b 4 a 4, b 3 a -3, b -4 a -4, b -3

These pairs of values satisfy the given equations.