Exercise 7 requires us to solve the equation
4 to the power of x plus 2 to the power of x+2 minus 32 equals 0.
Now, with some simple manipulation, we'll soon be able to see
that this equation is, in fact, a quadratic equation in disguise.
So we know that 4 is equivalent to 2 squared,
so this is 2 squared to the power of x.
And we also know that 2 to the power of x+2 could be broken up,
because when we multiply powers together, we add the indices,
and so 2 to the power of x+2 could be split up
as 2 to the power of x multiplied by 2 to the power of 2.
So now we have 2 to the power of 2x plus ...
Now, 2 squared is 4, so we've got 4 times 2 to the power of x,
and minus 32, which equals 0.
Now, 2 to the power of 2x could be thought of
as 2 to the power of x, all squared ...
... plus 4 times 2 to the power of x minus 32 equals zero.
So now we're starting to see the quadratic shape arise.
So we've got 2 to the power of x all squared
plus 4 times 2 to the power of x minus 32.
It's possible to solve from here, but for many people,
it's easier to see the quadratic by making a substitution.
So if we were to let u equal 2 to the power of x ...
... we should now be able to see ...
... we have a very straightforward quadratic equation.
We now have u squared plus four times u
minus 32 equals 0.
So to solve the quadratic equation, we'll attempt to factorise,
so we're looking for factors of –32 that sum to give +4.
And they are –4 and +8.
So we have u minus 4 and u plus 8.
Using the null factor law gives us the solutions
u equals 4 or –8.
But we weren't asked to solve for u. We're trying to solve for x.
So we need to replace the substitution that we made.
So we now have that u equals 4 or u equals –8.
So that is 2 to the power of x equals 4 or 2 to the power of x equals –8.
Considering the first of these two equations,
we attempt to express both sides of the equation
as a power with the same base,
so 2 to the power of x is equal to 2 to the power of 2,
which therefore means that x equals 2 is a solution to the equation.
In the second equation, we're looking at 2 to the power of x equals –8.
Now, there's no possible value of x that we could give
to make 2 to the power of x a negative number.
We could also think about the graph of 2 to the power of x,
which we know is an exponential graph that looks something like this.
And again we can see that that graph is never going to be equal
to a negative number.
So 2 to the power of x can never equal –8.
So therefore, there are no solutions from this part of the equation.
So x equals 2 is the only solution to our equation.