Solving for a^2 - ab - b^2 Using Given Values

In this article, we will explore how to solve the polynomial expression a2 - ab - b2 using the given values a x b 7 and ab 12. This example requires a step-by-step approach and showcases fundamental algebraic techniques including polynomial manipulation, substitution, and solving systems of equations using quadratics.

Step 1: Expressing the Polynomial

The goal is to find the value of a2 - ab - b2 using the given equations:

a b 7 a x b 12

First, we manipulate the expression a2 - ab - b2 into a more manageable form:

a2 - ab - b2 a2 b2 - ab

Step 2: Using the Identity

Next, we use the identity for the product of squares:

a2 b2 a ba - 2ab

Substituting the given value of a b 7, we get:

a2 b2 72 - 2ab

Step 3: Calculating Individual Terms

Now, we need to calculate a2 b2 and 2ab:

a2 b2 (a b)2 - 2ab a2 b2 72 - 2 x 12 49 - 24 25 2ab 2 x 12 24

Step 4: Substitution and Final Calculation

Substituting these values back into our manipulated expression:

a2 - ab - b2 (a2 b2) - ab

a2 - ab - b2 25 - 12 13

In conclusion, the value of a2 - ab - b2 is:

boxed{13}

Alternative Solutions

First Solution

1. From a b 7, we get a 7 - b.

2. Substitute b 12/a into a 7 - b:

3. Solving the quadratic equation a2 - 7a 12 0, we get a 3 or a 4. Thus, a 4 and b 3 (or vice versa).

4. Finally, a2 - ab - b2 42 - 4 x 3 - 32 16 - 12 - 9 13.

Second Solution

1. From a x b 12, we know ab^2 12a 49.

2. Using the identity a2 - ab - b2 ab^2 - 3ab:

3. Substituting ab 12, ab^2 49, we get:

4. a2 - ab - b2 49 - 3 x 12 49 - 36 13.

Third Solution

1. Given a x b 12 and a b 7, we solve the quadratic equation x2 - 7x 12 0.

2. Solving for a and b, we get a 3, b 4 or a 4, b 3.

3. Using a2 - ab - b2 13, we substitute and get:

4. 42 - 4 x 3 - 32 16 - 12 - 9 13.

Thus, the final solution is boxed{13}.