Section 4: Solving Quadratic in Form Equations

An equation that is Quadratic in Form is an equation that can be converted to quadratic form by making a substitution. For example, the equation x⁴ − 14x² + 49 = 0 can be converted to a quadratic equation by making the substitution y = x².

x⁴ − 14x² + 49 = 0      becomes      y² − 14y + 49 = 0

Once the equation is converted into quadratic form, the equation can be solved by factoring, completing the square, or by using the Quadratic Formula.

 y² − 14y + 49 = 0 can be factored into (y − 7)² = 0

This gives the solution for y:       y = 7

Now, we can back-substitute to get solution(s) for x. Recall that y = x².

  y = 7      becomes       x² = 7

Solve for x by applying the Square Root Property.

x = ± √ 7

Example 1. Solve the equation (x − 2)^2/3 + (x − 2)^1/3 − 2 = 0.

Solution

This equation can be converted to a quadratic equation by making the substitution y = (x − 2)^1/3.

[(x − 2)^1/3]²+ (x − 2)^1/3− 2 = 0     becomes      y² + y − 2 = 0

This equation can be solved by factoring.

  y² + y − 2 = 0 can be factored into (y − 1)(y + 2) = 0

Set the factors equal to zero and solve:

y − 1 = 0         or         y + 2 = 0

y = 1         or         y = −2

Back-substitute to get solution(s) for x. Recall that y = (x − 2)^1/3.

  y = 1      becomes      (x − 2)^1/3 = 1         and           y= −2     becomes      (x− 2)^1/3 = −2

Solve for x by cubing each side of each equation, and then adding 2.

For equations which include roots other than the square root, you want to remove the roots by (1) isolating the root term on one side of the equation, and (2) raising both sides of the equation to the appropriate power.