Video Transcript
Solve two times 𝑥 plus four
squared is equal to 162.
In this question, we want to find
the solutions to an equation. To do this, we want to isolate 𝑥
on one side of the equation by applying the same operations to both sides of the
equation.
We can first note that we can
divide both sides of the equation through by two to cancel the factor of two on the
left-hand side of the equation. On the left-hand side of the
equation, we are left with 𝑥 plus four all squared. And on the right-hand side of the
equation, we have 162 over two, which is equal to 81. To isolate 𝑥 on the left-hand side
of the equation, we can now note that we are squaring an expression for 𝑥 on the
left-hand side of the equation. We can reverse this operation by
taking square roots of both sides of the equation; however, we need to be
careful.
When we are solving an equation by
taking square roots, we need to be aware that there can be both positive and
negative solutions. For instance, the equation 𝑦
squared equals 81 has two solutions, 𝑦 equals nine and 𝑦 equals negative nine,
that is, the two numbers whose square is 81. Keeping this in mind, when we try
to solve an equation by taking square roots, we always need to consider both the
negative and positive root. This means that we have two cases,
one for each root. Either 𝑥 plus four is equal to
nine or 𝑥 plus four is equal to negative nine.
We can now solve each equation
separately by isolating 𝑥. For the first equation, we subtract
four from both sides of the equation to get that 𝑥 is equal to five. We can solve the second equation in
the same way. We subtract four from both sides of
the equation to obtain a second solution of 𝑥 equals negative 13. This is enough to answer the
question. However, we can check our answers
by substituting the values of 𝑥 back into the equation.
If we substitute 𝑥 equals five
into the left-hand side of the equation, we get two times five plus four
squared. We evaluate the sum inside the
parentheses to get two times nine squared. We then find that nine squared is
81 and doubling this value gives 162, which is the same as the right-hand side of
the given equation, verifying that 𝑥 equals five is a solution to the equation. In the same way, we can verify that
𝑥 equals negative 13 is a solution to the equation. Hence, if 𝑥 is a solution to the
equation, then either 𝑥 equals five or 𝑥 equals negative 13.
