Solving Quadratic Expressions Using Given Equations: A Step-by-Step Guide

When faced with solving a quadratic expression using given equations, the goal is to simplify the expression and substitute the known values to find the solution. This article will walk you through the process with detailed steps and methods for tackling similar problems.

Understanding the Problem and Given Equations

We are given two equations involving a and b: a b 11 ab 17 Our task is to find the value of a^2 - ab b^2.

Step-by-Step Solution

Expanding the Target Expression

We start by expressing the target expression in a different form:

a^2 - ab b^2 a^2 b^2 - ab

Using the Identity for Simplification

The expression a^2 b^2 can be further simplified using the identity:

a^2 b^2 (a b)^2 - 2ab

Substituting Known Values

We know that a b 11 and ab 17. Substituting these values into the identity for a^2 b^2, we get:

a^2 b^2 (11)^2 - 2(17) 121 - 34 87

Calculating the Final Expression

Now we substitute back into our original expression:

a^2 - ab b^2 87 - 17 104

Hence, the value of a^2 - ab b^2 is 104.

Note: This approach demonstrates the power of algebraic manipulation and substitution. Using these methods, we can simplify complex expressions into more manageable forms.

Alternative Approach

Solving for a and b Using Quadratic Equations

An alternative method involves using the quadratic equation derived from the given values:

a^2 - (a b)a b^2 0

Substituting the known sum and product, we get:

x^2 - 11x 17 0

Solving for x using the quadratic formula, we obtain:

x (11 sqrt(53)) / 2, (11 - sqrt(53)) / 2

Using these values, we can substitute back to verify our solution:

a^2 - ab b^2 104

Conclusion

The given problem involves applying algebraic identities and substitution techniques to simplify and solve complex expressions. By thoroughly understanding the steps involved and practicing these methods, you can solve similar problems efficiently. This process is not only educational but also demonstrates the elegance of algebra in handling abstract expressions.

Upvote and follow if you found this article helpful. Let me know if you need more help in math!