Can somebody explain to me on how I can do this problem?
Find all real solutions to this equation. x^2+x+ radical x^2+x-2=0.
What part is under the radical sign ?
Is it
x^2 + x + √(x^2+ x - 2) = 0 ?
Yes it is x^2+x+radical(x^2+x)-2=0.
Sure! To find the real solutions to the equation x^2 + x + sqrt(x^2 + x - 2) = 0, we can follow these steps:
Step 1: Rewrite the equation without the square root by isolating it on one side.
Start by subtracting x from both sides of the equation:
x^2 + sqrt(x^2 + x - 2) = -x
Step 2: Square both sides of the equation to eliminate the square root.
Take both sides of the equation and square them:
(x^2 + sqrt(x^2 + x - 2))^2 = (-x)^2
This simplifies to:
x^2 + 2x sqrt(x^2 + x - 2) + x^2 + x - 2 = x^2
Step 3: Simplify the equation.
Combine like terms and move everything to one side of the equation:
2x sqrt(x^2 + x - 2) + 2x - 2 = 0
Step 4: Factor out the common factor.
Factor out 2x from the equation:
2x(sqrt(x^2 + x - 2) + 1) - 2 = 0
Simplify further:
2x(sqrt(x^2 + x - 2) + 1) = 2
Divide both sides by 2:
x(sqrt(x^2 + x - 2) + 1) = 1
Step 5: Solve for x.
Divide both sides of the equation by (sqrt(x^2 + x - 2) + 1):
x = 1 / (sqrt(x^2 + x - 2) + 1)
Now that we have the equation x = 1 / (sqrt(x^2 + x - 2) + 1), we can find the real solutions by substituting different values of x and checking if they satisfy the equation. However, it is difficult to find an exact solution to this equation algebraically. In this case, we can use numerical methods such as graphing or numerical approximation methods to find real solutions to this equation.