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Solving Nonlinear Equations by Substitution

An equation that can be solved using this approach is said to be reducible to quadratic form.

 

Procedure — To Solve an Equation Reducible to Quadratic Form

Step 1 Write the equation in quadratic form.

Step 2 Use an appropriate “u” substitution.

Step 3 Solve the resulting equation.

Step 4 Substitute the original expression for u.

Step 5 Solve for the original variable.

 

The letter u is traditionally used when solving an equation by the substitution method.

You may use a different letter if you prefer.

 

Example

Solve for x: x4 - 21x2 - 35 = 65

Solution

Step 1 Write the equation in quadratic form.

Subtract 65 from both sides.

Write x4 as (x2)2.

Step 2 Use an appropriate “u” substitution.

Substitute u for x2.

x4 - 21x2 - 35

 

x4 - 21x2 - 100

(x2)2 - 21(x2)1 - 100

 

u2 - 21u - 100

= 65

 

= 0

= 0

 

= 0

Use the Zero Product Property.

Solve each equation for u.

Step 4 Substitute the original expression for u.

Step 5 Solve for the original variable.

 u + 4 = 0 or u - 25

 u = -4 or u

x2 = -4 or x2

 

= 0

= 25

= 25

 

Take the square root of both sides.

So, there are four solutions: -2i, +2i, -5, and +5. Needless to say, these are the same solutions that we would obtain using factoring.

Note:

The exponent of x4 is twice that of x2. So, we let u = x2.

Then, x4 = (x2)2 = u2.

 
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