*Please solve the rational expression.
x^2/x-4 - 7/x-4=0
To solve the rational expression (equation) x^2/(x-4) - 7/(x-4) = 0, you can follow these steps:
Step 1: Simplify the expression on the left-hand side.
To add or subtract rational expressions, you need a common denominator. In this case, the common denominator is (x-4). Rewrite the expression as follows:
x^2/(x-4) - 7/(x-4) = (x^2 - 7)/(x-4) = 0
Step 2: Set the numerator equal to zero.
The only way for the fraction to equal zero is if the numerator (x^2 - 7) is equal to zero. Set the numerator equal to zero:
x^2 - 7 = 0
Step 3: Solve the quadratic equation.
To solve the quadratic equation x^2 - 7 = 0, you can factor it or use the quadratic formula. In this case, since the equation cannot be easily factored, we will use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
For our quadratic equation, a = 1, b = 0, and c = -7. Plug these values into the quadratic formula:
x = (-0 ± √(0^2 - 4(1)(-7)))/(2(1))
x = (± √(0 + 28))/2
x = (± √(28))/2
x = ± √(28)/2
x = ± √(4 × 7)/2
x = ± (2 √7)/2
x = ± √7
Therefore, the solutions to the equation are x = √7 and x = -√7.
So, x = √7 and x = -√7 are the solutions to the original rational expression (equation) x^2/(x-4) - 7/(x-4) = 0.