Can someone help me foil this equation?
[�ã(2x+2) - �ã(x-3)][�ã(2x+2) - �ã(x-3)] = 4
Of course! I can help you with that. To foil this equation, we will expand and simplify the expression on the left side of the equation.
The expression you provided is:
√(2x+2) - √(x-3) * √(2x+2) - √(x-3) = 4
To simplify this equation, we can use the foil method. "FOIL" stands for First, Outer, Inner, Last, which is a technique for multiplying two binomials.
Let's go step by step:
First, we multiply the First terms from both binomials:
(√(2x+2)) * (√(2x+2)) = (√(2x+2))² = 2x + 2
Next, we multiply the Outer terms from both binomials:
(√(2x+2)) * (-√(x-3)) = -√((2x+2)(x-3))
Then, we multiply the Inner terms from both binomials:
(-√(x-3)) * (√(2x+2)) = -√((2x+2)(x-3))
Finally, we multiply the Last terms from both binomials:
(-√(x-3)) * (-√(x-3)) = (√(x-3))² = x - 3
Now, let's rewrite the equation with the FOILed expression:
(2x + 2 - √((2x+2)(x-3)) - √((2x+2)(x-3))) = 4
Now that we have expanded the equation, we can simplify it further. Let's combine like terms:
(2x + 2) - √((2x+2)(x-3)) - √((2x+2)(x-3)) = 4
We notice that we have two terms with the square root of the same expression. To combine these terms, we can factor out the common square root:
(2x + 2) - 2√((2x+2)(x-3)) = 4
Now, let's move all terms to one side of the equation to isolate the square root expression:
(2x + 2) - 4 = 2√((2x+2)(x-3))
Simplifying further, we have:
2x - 2 = 2√((2x+2)(x-3))
Now, divide both sides of the equation by 2 to solve for the square root:
x - 1 = √((2x+2)(x-3))
Now, square both sides to eliminate the square root:
(x - 1)² = ((2x+2)(x-3))
Expanding and simplifying further:
x² - 2x + 1 = 2x² - 2x - 6
Now, gather all terms on one side of the equation to solve for x:
x² - 2x + 1 - 2x² + 2x + 6 = 0
Combine like terms:
-x² + 7 = 0
To solve this quadratic equation, we can either factor or use the quadratic formula.
Factoring the equation:
(-x + √7)(x + √7) = 0
Setting each factor equal to zero:
-x + √7 = 0
x + √7 = 0
Solving for x in each equation:
-x = -√7
x = -√7
x = √7
So the solutions to the equation are x = -√7 and x = √7.
Please note that this is a lengthy process, and due to the complexity of the equation, there is always a possibility of errors in the calculations. I would recommend double-checking the solution or using a graphing calculator for confirmation.