Solve for x: x^4-15X^2+14=0
How would I solve this problem? Could someone please show me the steps?
Enter the equation into a graphing calculator. Then use the zero function under the 2nd Trace menu. Find the zeros.
Or, notice that it's just a quadratic in x^2:
(x^2 - 14)(x^2 - 1) = 0
x = ±√14 or ±1
To solve the equation x^4 - 15x^2 + 14 = 0, you can use a substitution method to simplify the equation. Let's introduce a new variable, u, which is equal to x^2. This allows us to rewrite the equation as a quadratic equation in terms of u:
u^2 - 15u + 14 = 0
Now we can solve for u using the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
For the equation u^2 - 15u + 14 = 0, a = 1, b = -15, and c = 14. Substituting these values into the quadratic formula, we have:
u = (-(-15) ± √((-15)^2 - 4(1)(14))) / (2(1))
Simplifying further, we get:
u = (15 ± √(225 - 56)) / 2
u = (15 ± √169) / 2
u = (15 ± 13) / 2
This gives us two possible values for u:
Case 1: u = (15 + 13) / 2 = 14
Case 2: u = (15 - 13) / 2 = 1
Now we can substitute back into the original equation to solve for x:
For Case 1: x^2 = 14
Taking the square root of both sides, we get:
x = ±√14
For Case 2: x^2 = 1
Taking the square root of both sides, we get:
x = ±1
So the possible solutions to the equation x^4 - 15x^2 + 14 = 0 are:
x = √14, -√14, 1, -1.