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We need to find all real solutions to this equation.
Quadratic equations and powers are used throughout ...
school and university, so they are important to ...
master. I'll give you 5 more seconds to ...
get your head around.
Right, let's get started!
To solve this equation,  we need to be familiar with ...
the quadratic formula. If you need to practise,
check out my previous videos on this topic.
We also need to remind ourselves that any number to ...
the power of zero is 1, except zero.
0^0 is undefined. And we also need to realise ...
that 1 raised to any power will equal 1.
So, we need solving two ...
equations. Either the exponent is 0 or the ...
base is 1. First,
we find the values of x that make the exponent zero.
We use the quadratic formula with a=1,
b=-3 and c=4. We end up having to deal with ...
the square root of -7, which is not a real number.
x^2-3x+4=0 does not have any real solutions.
You can double check it by drawing the graph of ...
y=x^2-3x+4 which does not cross the x-axis.
Second, we find the values of x that ...
make the base 1. x=Sqrt(2) and x=-Sqrt(2).
Is that all? Not really!.
We always need to reason carefully that we have ...
considered all the possibilities. We can also make the base ...
-1 as long as the exponent is even,
since (−1) raised to an even number will return 1.  3−x^2= ...
−1 has solutions x=±2. In this case the exponent ...
becomes 8±6, which in either case is an ...
even number. So 2 and -2 are valid solutions.
The four real solutions to the equation are Sqrt(2),
-Sqrt(2), 2 and -2.
Powers, roots and quadratics link ...
together very nicely when complex numbers are ...
considered. So,
as an extension, you could try to find the ...
complex solutions and share your answers on the ...
comments below.
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