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.

Also Read:

Also Read : Solve revenue and distance applications of quadratic equations

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.

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