Solving for the Value of ( ab ) Given ( a times b 7 ) and ( a^2 times b^2 25 )

In this article, we will demonstrate several methods to solve for the product ( ab ) given the conditions ( a times b 7 ) and ( a^2 times b^2 25 ). We will explore a step-by-step approach to understanding and solving these types of algebraic problems using various algebraic techniques, including substitution, algebraic manipulation, and the quadratic formula.

Introduction to the Problem

Given the equations ( a times b 7 ) and ( a^2 times b^2 25 ), our goal is to find the value of ( ab ). Let's explore different methods to reach this solution.

Method 1: Direct Algebraic Manipulation

First, let's use direct algebraic manipulation to solve for ( ab ).

We start with the given equations: ( a times b 7 ) ( a^2 times b^2 25 )

From the second equation, we can rewrite it as:

( (a times b)^2 25 )

Since ( a times b 7 ), we substitute:

( (7)^2 25 )

This simplifies to ( 7^2 25 ), which means:

( 49 25 )

Clearly, we need to manipulate the equation differently. We can infer:

( (a times b)^2 25 )

Therefore:

( a times b pm 5 )

However, we already know from ( a times b 7 ) that:

( a times b 7 )

Thus, the correct value is:

( a times b 12 )

Method 2: Using the Quadratic Formula

Let's use the quadratic formula to solve for ( a ) and ( b ).

We start by expressing ( a ) in terms of ( b ) from the first equation:

( a frac{7}{b} )

Substitute ( a ) into the second equation:

( left(frac{7}{b}right)^2 times b^2 25 )

This simplifies to:

( frac{49}{b^2} times b^2 25 )

( 49 25 )

Again, this confirms our previous calculation. To find the specific values, we solve:

( a^2 - 7a 12 0 )

Applying the quadratic formula ( a frac{-b pm sqrt{b^2 - 4ac}}{2a} ):

( a frac{7 pm sqrt{49 - 48}}{2} )

( a frac{7 pm sqrt{1}}{2} )

( a frac{7 1}{2} ) or ( a frac{7 - 1}{2} )

( a 4 ) or ( a 3 )

Correspondingly, ( b 3 ) or ( b 4 ). Therefore, ( ab 4 times 3 12 ) or ( ab 3 times 4 12 ).

Conclusion

In conclusion, through various algebraic manipulations and the application of the quadratic formula, we find that the value of ( ab ) is consistently 12 given the conditions ( a times b 7 ) and ( a^2 times b^2 25 ).

Understanding these methods will help in solving similar algebraic problems involving equations and quadratic expressions. The key takeaway is the importance of simplifying and substituting correctly to derive the solution.