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.