Video Transcript
Find the solution set for 𝑥 to the
fourth power minus 13𝑥 squared plus 36 is equal to zero.
The equation we’ve been asked to
solve is of degree four, and hence it is a quartic equation. However, by the laws of exponents
or indices, we know that 𝑥 to the fourth power is equal to 𝑥 squared squared. This means that we have a
quadratic-like equation. This can be rewritten as 𝑥 squared
squared minus 13 multiplied by 𝑥 squared plus 36 is equal to zero. We can then introduce a new
variable 𝑦 such that 𝑦 is equal to 𝑥 squared, and our equation becomes 𝑦 squared
minus 13𝑦 plus 36 equals zero.
We can now solve this equation for
𝑦 and then subsequently solve for 𝑥. The quadratic expression factors
into 𝑦 minus four multiplied by 𝑦 minus nine, as negative four and negative nine
have a sum of negative 13 and a product of 36. We therefore have two possible
solutions: 𝑦 minus four equals zero or 𝑦 minus nine equals zero. And hence 𝑦 is equal to four or 𝑦
is equal to nine.
Recalling our substitution and
solving for 𝑥, we have 𝑥 squared is equal to four and 𝑥 squared is equal to
nine. Both of these equations have two
real roots: 𝑥 is equal to positive or negative root four and 𝑥 is equal to
positive and negative root nine. Since the square root of four is
two and the square root of nine is three, we have 𝑥 is equal to positive or
negative two and positive or negative three. The solution set for 𝑥 to the
fourth power minus 13𝑥 squared plus 36 equals zero has four values: two, negative
two, three, and negative three.
