# 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 $0=ax^2+bx+c$, 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: $0=ax^m+bx^n+c$ 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!.

Example:

Solve the solution $x^4-13x^2+36=0$

Solution:

Quadratic form, one exponent, 4, double the other, 2

$y=x^2$

New variable equal to the variable with smaller exponent

$y^2=x^4$

Substitute y for x2 and y2 for x4 .

$y^2-13y+36=0$

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

$x^2=9$ or $x^2=4$

$x=\pm \sqrt{9}$ or $x=\pm \sqrt{4}$

$x=\pm 3$ or $x=\pm 2$

Our solution is $x=\pm 2,\pm 3$.

×