Final answer:
To find the value of a² + b², substitute the given values into the equation and use substitution or elimination to solve for a and b. However, in this case, there are no real solutions for a and b.
Step-by-step explanation:
To find the value of a² + b², we can substitute the given values into the equation. The second equation given, a² + b² = 6, tells us that the sum of the squares of a and b is equal to 6. To solve for a and b, we can use the third equation a - b = 4. We can use substitution or elimination to find the values of a and b, and then calculate a² + b² using these values.
Let's use substitution to solve for a and b. From the third equation, we can rearrange it to get a = b + 4. Substituting this expression into the second equation, we get (b + 4)² + b² = 6. Expanding the square, we get b² + 8b + 16 + b² = 6. Combining like terms, we have 2b² + 8b + 10 = 0. Now, we can solve this quadratic equation for b using the quadratic formula.
Using the quadratic formula, we have b = (-8 ± √(8² - 4(2)(10))) / (2(2)). Simplifying further, we get b = (-8 ± √(64 - 80)) / 4, which becomes b = (-8 ± √(-16)) / 4. Since the discriminant (√(-16)) is negative, there are no real solutions for b. Therefore, there are no real values of a and b that satisfy the given system of equations.