find all exact real solutions of the equation.
x^4-2x^2-24=0
Use the substitution
y=x²
y²-2y-24=0
factorize
(y-6)(y+4)=0
y=6, y=-4
x²=6 ... (1)
x²=-4 ... (2)
(1) gives x=±√6 (real solutions)
(2) gives x=±2i (complex solutions, discarded)
X²+y²+2y-24=0
To find all the exact real solutions of the equation x^4 - 2x^2 - 24 = 0, we can use factoring and solving methods. Here's how:
Step 1: Rewrite the equation in a quadratic form.
Let's substitute x^2 = a. The equation becomes a^2 - 2a - 24 = 0.
Step 2: Factor the quadratic equation.
We can factor the quadratic equation as (a - 6)(a + 4) = 0.
Step 3: Solve for x.
Since a = x^2, we can solve for x by taking the square root of each side. We get x^2 = 6 or x^2 = -4.
Step 4: Find the square roots of both sides.
For x^2 = 6, taking the square root of both sides gives x = √6 or x = -√6.
For x^2 = -4, since taking the square root of a negative number gives imaginary solutions, there are no real solutions.
Therefore, the exact real solutions of the equation x^4 - 2x^2 - 24 = 0 are x = √6 and x = -√6.