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Exercise 7 requires us
to solve the equation

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4 to the power of x plus 2 to
the power of x+2 minus 32 equals 0.

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Now, with some simple manipulation,
we'll soon be able to see

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that this equation is, in fact,
a quadratic equation in disguise.

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So we know that 4 is equivalent
to 2 squared,

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so this is 2 squared
to the power of x.

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And we also know that 2 to
the power of x+2 could be broken up,

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because when we multiply powers
together, we add the indices,

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and so 2 to the power of x+2
could be split up

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as 2 to the power of x
multiplied by 2 to the power of 2.

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So now we have
2 to the power of 2x plus ...

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Now, 2 squared is 4, so we've got
4 times 2 to the power of x,

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and minus 32, which equals 0.

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Now, 2 to the power of 2x
could be thought of

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as 2 to the power of x,
all squared ...

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... plus 4 times 2 to the power of x
minus 32 equals zero.

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So now we're starting to see
the quadratic shape arise.

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So we've got
2 to the power of x all squared

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plus 4 times 2 to the power of x
minus 32.

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It's possible to solve from here,
but for many people,

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it's easier to see the quadratic
by making a substitution.

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So if we were to let u equal
2 to the power of x ...

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... we should now be able to see ...

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... we have a very straightforward
quadratic equation.

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We now have u squared
plus four times u

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minus 32 equals 0.

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So to solve the quadratic equation,
we'll attempt to factorise,

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so we're looking for factors of –32
that sum to give +4.

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And they are –4 and +8.

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So we have u minus 4
and u plus 8.

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Using the null factor law
gives us the solutions

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u equals 4 or –8.

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But we weren't asked to solve for u.
We're trying to solve for x.

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So we need to replace
the substitution that we made.

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So we now have
that u equals 4 or u equals –8.

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So that is 2 to the power of x equals
4 or 2 to the power of x equals –8.

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Considering the first
of these two equations,

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we attempt to express
both sides of the equation

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as a power with the same base,

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so 2 to the power of x
is equal to 2 to the power of 2,

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which therefore means that x equals 2
is a solution to the equation.

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In the second equation, we're looking
at 2 to the power of x equals –8.

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Now, there's no possible value of x
that we could give

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to make 2 to the power of x
a negative number.

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We could also think about the graph
of 2 to the power of x,

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which we know is an exponential graph
that looks something like this.

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And again we can see that that graph
is never going to be equal

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to a negative number.

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So 2 to the power of x
can never equal –8.

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So therefore, there are no solutions
from this part of the equation.

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So x equals 2 is the only solution
to our equation.