Video Transcript
Solve 36π₯ to the power four minus
97π₯ squared plus 36 equals zero.
Notice the presence of a fourth
power here. This tells us that this is a
fourth-degree, or quartic, equation. There does exist an algorithm
analogous to the quadratic formula for solving quartic equations, but it is
extremely complicated and difficult. Rather, when presented with
higher-degree equations such as this, one should be on the lookout for some kind of
trick to simplify the situation.
In particular, one thing to look
out for is situations in which all of the terms have even degree. That is the case here. In such cases, one can use the laws
of exponents to rewrite the equation in terms of π₯ squared. Making the substitution π’ equals
π₯ squared, we have reduced the problem to a quadratic, which we can solve. You can use the quadratic formula
or stare at factor pairs of 36 until you see how it factors. Either way, we find the two
solutions π’ equals nine over four and π’ equals four over nine.
Beware though! To solve the original equation, we
need to find the values of π₯ which make it true, and we havenβt done that yet. Since we made the substitution π’
equals π₯ squared, we have that π₯ equals the square root of π’. Thus, we have the four solutions
root nine over four, negative root nine over four, root four over nine, and negative
root four over nine. Observe that these are all ratios
of perfect squares and simplify down to the solutions three over two, negative three
over two, two over three, and negative two over three.
