solve the equation
5 - √x = 2x^2 - 3
To solve the equation 5 - √x = 2x^2 - 3, we need to isolate x on one side of the equation. Here's how you can do that:
Step 1: Start by isolating the square root term on one side of the equation. Add √x to both sides:
5 - √x + √x = 2x^2 - 3 + √x
Simplifying this equation gives us:
5 = 2x^2 - 3 + √x
Step 2: Next, move all terms without square roots to one side of the equation and bring the terms with square roots together on the other side. This will allow us to isolate x:
√x = 2x^2 - 8
Step 3: Square both sides of the equation to eliminate the square root:
(√x)^2 = (2x^2 - 8)^2
Simplifying this equation gives us:
x = (2x^2 - 8)^2
Step 4: Expand and simplify the equation further by squaring the binomial:
x = 4x^4 - 32x^2 + 64
At this point, we have transformed the original equation into a fourth-degree polynomial equation. To solve for x, we can factor, use the Rational Root Theorem, or approximate using numerical methods.
Note that finding exact solutions to fourth-degree polynomial equations can be quite complex and might require advanced techniques.