Solve equations that are quadratic in form by substitution to create a quadratic equation
We have seen three different ways to solve quadratics: factoring, completing the square, and the quadratic formula. A quadratic is any equation of the form , however, we can use the skills learned to solve quadratics to solve problems with higher (or sometimes lower) powers if the equation is in what is called quadratic form.
Quadratic form: where m = 2n.
An equation is in quadratic form if one of the exponents on a variable is double the exponent on the same variable somewhere else in the equation. If this is the case we can create a new variable, set it equal to the variable with smallest exponent. When we substitute this into the equation we will have a quadratic equation we can solve.
World View Note: Arab mathematicians around the year 1000 were the first to use this method!.
Solve the solution
Quadratic form, one exponent, 4, double the other, 2
New variable equal to the variable with smaller exponent
Substitute y for x2 and y2 for x4 .
Solve. We can solve this equation by factoring
(y – 9)(y – 4) = 0
Set each factor equal to zero
y – 9 = 0 or y – 4 = 0
y = 9 or y = 4
Substitute values for y
Our solution is .