Solving ( a^2 - b^2 12 ) and Finding ( ab ): A Comprehensive Guide for SEO

Solving Algebraic Equations: ( a^2 - b^2 12 )

When faced with the equation ( a^2 - b^2 12 ), one can use the difference of squares to factor and find the solution for ( ab ). The difference of squares formula helps us express the equation as:

[begin{align*}quad a^2 - b^2 (a - b)(a b) [5pt]quad (a - b)(a b) 12end{align*}]

We introduce new variables for simplification:

[x a - b quad text{and} quad y a b]

Thus, we have:

[(x)(y) 12]

Expressing (a) and (b) in terms of (x) and (y), we get:

[a frac{x y}{2} quad text{and} quad b frac{y - x}{2}]

Now, we find (ab):

[ab left(frac{x y}{2}right)left(frac{y - x}{2}right) frac{(x y)(y - x)}{4} frac{y^2 - x^2}{4}]

Given that (xy 12 ), we can use the integer factor pairs of 12 to find possible values for (ab):

[begin{align*}1 times 12 quad Rightarrow quad (a, b) left(frac{1 12}{2}, frac{12 - 1}{2}right) left(frac{13}{2}, frac{11}{2}right) quad Rightarrow quad ab frac{13 cdot 11}{4} frac{143}{4} [5pt]2 times 6 quad Rightarrow quad (a, b) left(frac{2 6}{2}, frac{6 - 2}{2}right) (4, 2) quad Rightarrow quad ab 8 [5pt]3 times 4 quad Rightarrow quad (a, b) left(frac{3 4}{2}, frac{4 - 3}{2}right) left(frac{7}{2}, frac{1}{2}right) quad Rightarrow quad ab frac{7}{4} [5pt]4 times 3 quad Rightarrow quad (a, b) left(frac{4 3}{2}, frac{3 - 4}{2}right) left(frac{7}{2}, -frac{1}{2}right) quad Rightarrow quad ab -frac{7}{4} [5pt]6 times 2 quad Rightarrow quad (a, b) left(frac{6 2}{2}, frac{2 - 6}{2}right) (4, -2) quad Rightarrow quad ab -8 [5pt]12 times 1 quad Rightarrow quad (a, b) left(frac{1 12}{2}, frac{12 - 1}{2}right) left(frac{13}{2}, frac{11}{2}right) quad Rightarrow quad ab -frac{143}{4}end{align*}]

Given the factor pairs, the possible values for (ab) are:

[frac{143}{4}, 8, frac{7}{4}, -frac{7}{4}, -8, -frac{143}{4}]

The positive solution is (ab 8).

Alternative Methods and Considerations

Another straightforward method to solve (a^2 - b^2 12) is to assume specific values for (a) and (b), such as (a 4) and (b 2). This gives:

[4^2 - 2^2 16 - 4 12 quad Rightarrow quad ab 4 cdot 2 8]

This method shows that (ab) can be found to be (8) with specific values of (a) and (b).

Mathematical Functions on Quora

Quora does support the editing of mathematical functions. You can use LaTeX or other math modes to input complex equations and formulas directly. If you need to learn more about how to use math functions on Quora, you can refer to the official guidelines and examples provided in the Quora Help Center. An introduction to beautiful math on Quora can be found here:

By following these resources, you can enhance the presentation of your mathematical problems and solutions on Quora.