Can someone help me solve this type of equation.
[SQRT(x + 3)] - [SQRT(x - 2)] =1
. Also. My teacher tells me I keep getting it wrong because after I square both sides, I am supposed to foil the "lefthand side of the equation". IF anyone knows what I am talking about and can help me, I would really appreciate it.
If you square the left side of the equation (and the right also), you get
x + 3 - 2 sqrt[(x-2)(x+3)] + x-2 = 1
That is where the "FOIL" is done, but you still have more to do. Combine terms and you get
2x - 2 sqrt[(x-2)(x+3)] = 0
Divide both sides by two, move the
-sqrt[(x-2)(x+3)] to the right side, and square both sides again.
x^2 = (x-2)(x+3)] = x^2 + x -6
Cancel out the x^2 terms
x - 6 = 0
x = 6
Plug it into your first equation and verify that it is a solution. it is
To solve the equation [√(x + 3)] - [√(x - 2)] = 1, let's break it down step by step.
Step 1: Isolate one of the radicals.
To start, let's move the radical on the right side of the equation to the other side by adding [√(x - 2)] to both sides:
[√(x + 3)] = 1 + [√(x - 2)]
Step 2: Square both sides.
By squaring both sides of the equation, we eliminate the square root and arrive at a solvable equation. Applying the square operation to both sides gives:
[√(x + 3)]^2 = (1 + [√(x - 2)])^2
Step 3: Simplify both sides.
On the left side, the square root and the square cancel each other out, leaving just (x + 3). On the right side, we need to "foil" (multiply the terms) in the expression (1 + [√(x - 2)])^2:
(x + 3) = (1 + [√(x - 2)]) * (1 + [√(x - 2)])
(x + 3) = 1 + [√(x - 2)] + [√(x - 2)] + [(√(x - 2))^2]
Step 4: Simplify further and solve.
In the expression on the right side, [√(x - 2)] + [√(x - 2)] is equal to 2√(x - 2). Additionally, [(√(x - 2))^2] is simply (x - 2). After combining like terms and simplifying, we have:
x + 3 = 1 + 2√(x - 2) + x - 2
Step 5: Further simplify and solve for x.
Let's simplify the equation by canceling out the x terms on both sides:
x + 3 = 1 + 2√(x - 2) + x - 2
3 = 1 + 2√(x - 2)
Next, let's isolate the square root term by subtracting 1 from both sides:
3 - 1 = 1 + 2√(x - 2) - 1
2 = 2√(x - 2)
Now, divide both sides by 2 to solve for the square root term:
2/2 = (2√(x - 2))/2
1 = √(x - 2)
Step 6: Square both sides to eliminate the square root.
By squaring both sides again, we can simplify the equation further:
(1)^2 = (√(x - 2))^2
1 = x - 2
Step 7: Solve for x.
Finally, by adding 2 to both sides of the equation, we can solve for x:
1 + 2 = x - 2 + 2
3 = x
Therefore, the solution to the equation [√(x + 3)] - [√(x - 2)] = 1 is x = 3.