Solving Mathematical Equations: a - b -8 and ab 3

Mathematics is a powerful tool for solving various real-world problems, from engineering to finance. One common challenge in algebra is solving systems of equations to find the values of variables. In this article, we will explore a particular problem involving two equations: a - b -8 and ab 3. We will use various algebraic methods to find the value of a^2 - b^2.

Identifying and Using Identities

To start, letrsquo;s use the identity a^2 - b^2 (a - b)^2 - 2ab.

Given that a - b -8 and ab 3, we can substitute these values into the identity mentioned above.

Step 1: Calculate (a - b)^2

We know that a - b -8, so:

((a - b)^2 (-8)^2 64)

Step 2: Calculate 2ab

We also know that ab 3, so:

(2ab 2 cdot 3 6)

Step 3: Combine the Values

Now, we use the identity to find a^2 - b^2:

(a^2 - b^2 (a - b)^2 - 2ab 64 - 6 58)

Using Quadratic Equations

A second approach involves solving the system using a quadratic equation. Letrsquo;s revisit the problem using this method.

Step 1: Express a in terms of b

From the first equation, a - b -8, we can express a in terms of b:

(a b - 8)

Step 2: Substitute into the Second Equation

Now, substitute a b - 8 into the second equation ab 3:

((b - 8) cdot b 3)

This simplifies to:

(b^2 - 8b - 3 0)

Step 3: Solve the Quadratic Equation

Using the quadratic formula x frac{-b pm sqrt{b^2 - 4ac}}{2a}, where A 1, B -8, and C -3:

(b frac{8 pm sqrt{(-8)^2 - 4 cdot 1 cdot (-3)}}{2 cdot 1} frac{8 pm sqrt{64 12}}{2} frac{8 pm sqrt{76}}{2} 4 pm sqrt{19})

So, b 4 sqrt{19} or b 4 - sqrt{19}.

Step 4: Find Corresponding Values of a

For each value of b found, find the corresponding value of a:

If b 4 sqrt{19}, then:

(a b - 8 (4 sqrt{19}) - 8 -4 sqrt{19})

If b 4 - sqrt{19}, then:

(a b - 8 (4 - sqrt{19}) - 8 -4 - sqrt{19})

Step 5: Calculate a^2 - b^2

Using the values of a and b, we calculate a^2 - b^2:

For b 4 sqrt{19} and a -4 sqrt{19}:

(a^2 - b^2 (-4 sqrt{19})^2 - (4 sqrt{19})^2 (16 - 8sqrt{19} 19) - (16 8sqrt{19} 19) 35 - 35 70)

For b 4 - sqrt{19} and a -4 - sqrt{19}:

(a^2 - b^2 (-4 - sqrt{19})^2 - (4 - sqrt{19})^2 (16 8sqrt{19} 19) - (16 - 8sqrt{19} 19) 35 - 35 70)

In both cases, a^2 - b^2 70.