Show all work to factor x4 17x2 + 16 completely. from brainly.com
Introduction
Solving a fourth-degree equation can be challenging, but there are techniques that can simplify the process. One such method is the substitution method, where a substitution is made to transform the equation into a simpler form. In this article, we will use the substitution method to solve the equation x4 + 17x2 + 16 = 0.
Step 1: Let u = x2
The first step in the substitution method is to make a substitution that simplifies the equation. In this case, we can let u = x2. This transforms the equation into u2 + 17u + 16 = 0.
Why does this work?
This substitution works because x4 can be expressed as (x2)2 and x2 can be substituted with u. This simplifies the equation and makes it easier to solve.
Step 2: Solve for u
Now that we have transformed the equation, we can solve for u using the quadratic formula. The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a In this case, a = 1, b = 17, and c = 16. Substituting these values into the formula, we get: u = (-17 ± √(17^2 - 4(1)(16))) / 2(1) Simplifying this expression, we get: u = (-17 ± √225) / 2 u1 = -8 and u2 = -9
What do u1 and u2 mean?
u1 and u2 are the solutions for u. But we need to remember that u = x2. Therefore, we need to find the values of x that correspond to u1 and u2.
Step 3: Find the values of x
To find the values of x, we need to take the square root of u1 and u2. However, we need to be careful with the signs of the square roots.
Case 1: u1 = -8
If u1 = -8, then x2 = -8. However, this is not possible since the square of any real number cannot be negative. Therefore, there are no real solutions for u1.
Case 2: u2 = -9
If u2 = -9, then x2 = -9. Taking the square root of both sides, we get: x = ±√(-9) Since the square root of a negative number is an imaginary number, the solutions for x are complex numbers.
Step 4: Write the solutions
The solutions for the original equation x4 + 17x2 + 16 = 0 are: x = ±√(-9) or x = ±3i
Conclusion
The substitution method is a useful technique for solving equations. By making a substitution that simplifies the equation, we can transform a complex equation into a simpler form that can be solved more easily. In this case, we used the substitution u = x2 to transform the equation x4 + 17x2 + 16 = 0 into the quadratic equation u2 + 17u + 16 = 0. By solving for u using the quadratic formula, we found the values of u that correspond to the solutions for x. Finally, we took the square root of these values to find the solutions for x.
Share :
Post a Comment
for "Solving X<Sup>4</Sup> + 17X<Sup>2</Sup> + 16 = 0 Using Substitution
Method"
Post a Comment for "Solving X<Sup>4</Sup> + 17X<Sup>2</Sup> + 16 = 0 Using Substitution Method"