Video Transcript
Solve four 𝑥 cubed minus four 𝑥
squared minus 15𝑥 equals zero.
Generally, cubic equations are
quite hard to solve. However, notice that this
particular cubic has no constant term. In other words, each term has a
factor of 𝑥. We can immediately factor this
common 𝑥 out. One way for this product to be zero
is the factor of 𝑥 being zero. This gives us one solution of 𝑥
equals zero. The other solutions come from the
roots of the quadratic factor four 𝑥 squared minus four 𝑥 minus 15.
Let’s find the roots of this
quadratic by factoring it. We can do this by inspection. Observe that four equals two times
two and negative 15 equals three times negative five. Observe also that two times three
minus two times five equals negative four.
So our quadratic factors as two 𝑥
plus three times two 𝑥 minus five. This leads to the pair of solutions
𝑥 equals negative three over two and 𝑥 equals five over two. The solution set is zero, negative
three over two, and five over two.
