Answer: The equation that is equivalent to X^2 - 6x = 8 is X = 3 ± √17.
Explanation:
To find an equation equivalent to the given equation X^2 - 6x = 8, we can manipulate the equation using algebraic operations.
1. Move the constant term to the other side of the equation:
X^2 - 6x - 8 = 0
2. To find the equation that is equivalent to this quadratic equation, we can either factor it or use the quadratic formula.
Option 1: Factoring
By factoring the quadratic equation, we can express it as the product of two binomials:
(X - a)(X - b) = 0
To find a and b, we need to determine two numbers whose product is -8 and whose sum is -6 (the coefficient of x).
In this case, the numbers are -2 and -4:
(X - 2)(X - 4) = 0
Therefore, the equation that is equivalent to X^2 - 6x = 8 is (X - 2)(X - 4) = 0.
Option 2: Quadratic formula
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
X = (-b ± √(b^2 - 4ac)) / 2a
For the given equation X^2 - 6x - 8 = 0, a = 1, b = -6, and c = -8.
Plugging these values into the quadratic formula, we get:
X = (-(-6) ± √((-6)^2 - 4(1)(-8))) / (2(1))
Simplifying further:
X = (6 ± √(36 + 32)) / 2
X = (6 ± √68) / 2
X = (6 ± 2√17) / 2
X = 3 ± √17
Both (X - 2)(X - 4) = 0 and X = 3 ± √17 are equivalent equations to X^2 - 6x = 8.