What is the solution set of x 4 9 x – What is the solution set of x^4 – 9x? This question delves into the realm of algebra, where we seek to uncover the values of x that satisfy a given equation. As we embark on this mathematical journey, we will explore the concept of solution sets, isolate the variable x, factor the equation, identify its roots, and construct the solution set, unraveling the mysteries that lie within this algebraic equation.
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Definition of the Solution Set: What Is The Solution Set Of X 4 9 X
In mathematics, the solution set of an equation is the set of all values of the variable that make the equation true.
For example, the solution set of the equation x+ 1 = 5 is 4, because 4 is the only value of xthat makes the equation true.
Types of Solution Sets
Solution sets can be classified into three types:
- Finite solution setscontain a finite number of elements.
- Infinite solution setscontain an infinite number of elements.
- Empty solution setscontain no elements.
Isolating the Variable
To solve the equation x^4 – 9x = 0, we need to isolate the variable ‘x’ on one side of the equation and the constant on the other side.
Adding 9x to Both Sides
We start by adding 9x to both sides of the equation:
x^4
9x + 9x = 0 + 9x
x^4 = 9x
Taking the Fourth Root of Both Sides
Next, we take the fourth root of both sides of the equation:
√(x^4) = √(9x)
x^2 = ±3√x
Squaring Both Sides
We then square both sides of the equation to eliminate the square root:
(x^2)^2 = (±3√x)^2
x^4 = 9x
Solving for x, What is the solution set of x 4 9 x
Finally, we solve for x by dividing both sides of the equation by x:
x^4/x = 9x/x
x^3 = 9
x = 3
Therefore, the solution set of the equation x^4 – 9x = 0 is 3.
Factoring the Equation
In order to solve the equation x^4 – 9x = 0, we need to factor the left-hand side of the equation.
We can start by grouping the terms as follows:
(x^4 – 8x) – (x – 9) = 0
We can then factor each group as follows:
x(x^3 – 8) – (x – 9) = 0
We can then factor the x^3 – 8 term as follows:
x(x – 2)(x^2 + 2x + 4) – (x – 9) = 0
We can then simplify the equation as follows:
(x – 9)(x(x – 2)(x^2 + 2x + 4) – 1) = 0
Therefore, the solution set of the equation x^4 – 9x = 0 is 0, 2, 3, 9.
Identifying the Roots
The roots of an equation are the values of the variable that make the equation true. To find the roots of a factored equation, we set each factor equal to zero and solve for the variable.
Finding the Roots of a Factored Equation
- Factor the equation completely.
- Set each factor equal to zero.
- Solve each equation for the variable.
- The solutions to the equations are the roots of the original equation.
Constructing the Solution Set
The solution set of an equation consists of all the values of the variable that satisfy the equation. To construct the solution set, we use the roots of the equation, which are the values of the variable that make the equation equal to zero.
Steps to Construct the Solution Set
- Identify the roots of the equation.
- Write the solution set as the set of all values of the variable that are equal to the roots.
Example
Consider the equation x^2 – 9 = 0. The roots of this equation are x = 3 and x = -3. Therefore, the solution set is 3, -3.
Query Resolution
What is a solution set?
A solution set is a collection of values that satisfy a given equation or inequality.
How do you isolate the variable x?
To isolate the variable x, you perform algebraic operations on both sides of the equation until x is alone on one side.
What is factoring?
Factoring is a process of expressing an algebraic expression as a product of simpler expressions.
What are roots?
Roots are the values of the variable that make an equation true.
