What are the real or imaginary solutions of the polynomial equation?
x4 – 52x2 + 576 = 0
(1 point)
Responses
4, –4
4, –4
4, –6
4, –6
4, –4, 6, –6
4, –4, 6, –6
no solution
The real solutions of the polynomial equation x^4 – 52x^2 + 576 = 0 are 4 and -4.
To find the solutions of the polynomial equation x^4 – 52x^2 + 576 = 0, we can use the substitution method. Let's set x^2 = y.
Substituting x^2 = y in the equation, we get y^2 – 52y + 576 = 0.
Now, we can solve this quadratic equation for y.
Using the quadratic formula, y = (-(-52) ± √((-52)^2 - 4(1)(576))) / 2(1).
Simplifying, y = (52 ± √(2704 - 2304)) / 2.
Further simplifying, y = (52 ± √400) / 2.
Therefore, we have two possible values for y: y = (52 + 20) / 2 = 36 or y = (52 - 20) / 2 = 16.
Now, we can substitute these values back into the equation x^2 = y to find the solutions for x.
Taking the square root of both sides, we get x = ±√36 or ±√16.
Simplifying further, we have four possible solutions: x = ±6 or ±4.
Hence, the solutions to the polynomial equation x^4 – 52x^2 + 576 = 0 are 6, -6, 4, and -4.
To find the solutions of the polynomial equation x^4 – 52x^2 + 576 = 0, we can use factoring or the quadratic formula.
First, we can notice that this equation is in quadratic form with respect to x^2. Let's substitute y = x^2 and rewrite the equation in terms of y:
y^2 – 52y + 576 = 0
Now, we can solve this quadratic equation by factoring or using the quadratic formula.
Factoring:
To factor the quadratic equation, we need to find two numbers whose product is 576 and whose sum is -52. By trial and error, we find that the numbers are -4 and -48. Therefore, we can factor the equation as:
(y - 4)(y - 48) = 0
Now, we substitute back y = x^2:
(x^2 - 4)(x^2 - 48) = 0
Now, we can analyze the solutions for x:
For (x^2 - 4) = 0,
x^2 = 4
Taking the square root of both sides, we have:
x = ±2
For (x^2 - 48) = 0,
x^2 = 48
Taking the square root of both sides, we have:
x = ±√48 = ±4√3 (approximately ±6.928)
Hence, the real or imaginary solutions of the polynomial equation x^4 – 52x^2 + 576 = 0 are:
x = ±2 and x = ±4√3 (approximately ±6.928). This means the solution set is 4, -4, 6√3, and -6√3.