(x-6)%3D25%2F24.jpeg "(x-5)(x-6)=25/24")
Solving the Equation (x-5)(x-6) = 25/24
This article will guide you through the steps of solving the equation (x-5)(x-6) = 25/24.
1. Expanding the Equation
First, we need to expand the left side of the equation by multiplying the terms:
(x-5)(x-6) = x² - 6x - 5x + 30 = x² - 11x + 30
Now our equation becomes: x² - 11x + 30 = 25/24
2. Bringing All Terms to One Side
To solve this quadratic equation, we need to bring all the terms to one side:
x² - 11x + 30 - 25/24 = 0
3. Simplifying the Equation
To simplify, let's find a common denominator for 30 and 25/24:
x² - 11x + (720 - 25)/24 = 0
x² - 11x + 695/24 = 0
4. Solving the Quadratic Equation
Now we have a standard quadratic equation in the form ax² + bx + c = 0. We can solve this using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
In our equation, a = 1, b = -11, and c = 695/24. Let's substitute these values into the formula:
x = (11 ± √((-11)² - 4 * 1 * (695/24))) / (2 * 1)
x = (11 ± √(121 - 695/6)) / 2
x = (11 ± √(121 - 115.83)) / 2
x = (11 ± √5.17) / 2
5. Finding the Solutions
Finally, we can calculate the two possible solutions for x:
x₁ = (11 + √5.17) / 2 ≈ 6.29
x₂ = (11 - √5.17) / 2 ≈ 4.71
Therefore, the solutions to the equation (x-5)(x-6) = 25/24 are x ≈ 6.29 and x ≈ 4.71.