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 x 4 − 14 x 2 + 49 = 0 can be converted to a quadratic equation by making the substitution y = x ².

x 4 − 14 x 2 + 49 = 0 becomes y ² − 14 y + 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.

y ² − 14 y + 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 = x ².

y = 7 becomes x ² = 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 y ² + y − 2 = 0

This equation can be solved by factoring.

y ² + 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.

( x − 2) 1/3 = 1

Cube each side of the equation:

[( x − 2) 1/3 ] 3 = 1³

Simplify:

x − 2 = 1

Solve for x :

x = 3

( x − 2) 1/3 = −2

Cube each side of the equation:

[( x − 2) 1/3 ] 3 = (−2)³

Simplify:

x − 2 = −8

Solve for x :

x = −6

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.

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