by To solve the equation ( x^3 = 2x ), we’ll use algebraic methods. The equation can be simplified and factored to find the values of ( x ) that satisfy it. Here’s a detailed guide on how to do it:
Understanding the Equation
The given equation is ( x^3 = 2x ). This is a cubic equation (since the highest power of ( x ) is 3). Our goal is to find the values of ( x ) that make this equation true.
Step 1: Simplify the Equation
First, we bring all terms to one side of the equation to set it to zero. This is a standard approach for solving polynomial equations.
[ x^3 – 2x = 0 ]
Step 2: Factor Out Common Terms
Notice that each term in ( x^3 – 2x ) has an ( x ). So, we can factor out ( x ) from each term:
[ x(x^2 – 2) = 0 ]
Step 3: Apply the Zero Product Property
The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. Thus, we set each factor in the equation ( x(x^2 – 2) = 0 ) equal to zero:
- ( x = 0 )
- ( x^2 – 2 = 0 )
Step 4: Solve Each Factor
- First Factor: ( x = 0 ) is already solved.
- Second Factor: ( x^2 – 2 = 0 ) To solve ( x^2 – 2 = 0 ), add 2 to both sides: [ x^2 = 2 ] Now, take the square root of both sides. Remember to consider both the positive and negative roots: [ x = \pm\sqrt{2} ]
Step 5: Compile the Solutions
Now, compile the solutions from each factor. The solutions to ( x^3 = 2x ) are:
- ( x = 0 )
- ( x = \sqrt{2} )
- ( x = -\sqrt{2} )
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Conclusion
The cubic equation ( x^3 = 2x ) has three solutions: ( 0 ), ( \sqrt{2} ), and ( -\sqrt{2} ). These solutions can be verified by substituting them back into the original equation. This demonstrates the power of factoring and the Zero Product Property in solving polynomial equations.
Frequently Asked Questions (FAQ)
What does it mean to solve a cubic equation?
Solving a cubic equation involves finding the values of the variable (in this case, ( x )) that satisfy the equation. For the equation ( x^3 = 2x ), it means finding all values of ( x ) that make the equation true.
Why do we set the equation to zero?
Setting the equation to zero is a standard approach for solving polynomial equations, as it allows for factoring and applying the Zero Product Property.
Can all cubic equations be solved by this method?
While many cubic equations can be solved using similar methods of simplification and factoring, some require more complex approaches like the cubic formula or numerical methods.
What is the significance of the Zero Product Property in solving equations?
The Zero Product Property states that if a product of factors equals zero, then at least one of the factors must be zero. This is crucial in solving polynomial equations as it allows us to break down complex equations into simpler ones.
