Video Transcript
Find the solution set for two 𝑥
cubed is equal to 18𝑥.
The equation in this question is of
degree three and is, therefore, a cubic equation. We can solve the equation by
firstly subtracting 18𝑥 from both sides. This gives us two 𝑥 cubed minus
18𝑥 is equal to zero. Next, we observe that the two terms
on the left-hand side have a common factor of two 𝑥. The equation can therefore be
factored as two 𝑥 multiplied by 𝑥 squared minus nine is equal to zero.
We now have the product of a linear
term and a quadratic expression. The quadratic expression 𝑥 squared
minus nine is the difference of two squares, as it is written in the form 𝑎 squared
minus 𝑏 squared. We recall that this can be factored
into two linear expressions: 𝑎 plus 𝑏 and 𝑎 minus 𝑏. 𝑥 squared minus nine can therefore
be rewritten as 𝑥 plus three multiplied by 𝑥 minus three. And we have the equation two 𝑥
multiplied by 𝑥 plus three multiplied by 𝑥 minus three is equal to zero. We know that if the product of
three factors equals zero, at least one of the individual factors must be equal to
zero.
To find the solution set of the
equation, we need to solve the three equations two 𝑥 equals zero, 𝑥 plus three
equals zero, and 𝑥 minus three equals zero. This gives us three possible
solutions of 𝑥 equals zero, 𝑥 equals negative three, and 𝑥 equals three. The solution set for the equation
two 𝑥 cubed is equal to 18𝑥 is zero, negative three, and three. We could check each of these
solutions individually by substituting the values of 𝑥 back in to the original
equation.
