Solving the Equation (x7)² + (x2)² = 10
This article will guide you through solving the equation (x7)² + (x2)² = 10. This equation represents a quadratic equation in disguise, and we'll use algebraic manipulation and the quadratic formula to find the solutions.
Expanding and Simplifying

Expand the squares: (x7)² = (x7)(x7) = x²  14x + 49 (x2)² = (x2)(x2) = x²  4x + 4

Substitute the expanded terms back into the equation: x²  14x + 49 + x²  4x + 4 = 10

Combine like terms: 2x²  18x + 53 = 10

Move all terms to one side to form a standard quadratic equation: 2x²  18x + 43 = 0
Solving the Quadratic Equation
Now we have a standard quadratic equation in the form ax² + bx + c = 0, where a = 2, b = 18, and c = 43.
We can use the quadratic formula to solve for x:
x = (b ± √(b²  4ac)) / 2a

Substitute the values of a, b, and c: x = (18 ± √((18)²  4 * 2 * 43)) / (2 * 2)

Simplify: x = (18 ± √(324  344)) / 4 x = (18 ± √(20)) / 4 x = (18 ± 2√5i) / 4

Reduce the fraction: x = 9/2 ± √5i / 2
Therefore, the solutions to the equation (x7)² + (x2)² = 10 are x = 9/2 + √5i / 2 and x = 9/2  √5i / 2. These solutions are complex numbers, indicating that the equation does not have real roots.