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 Dependent Variable

 Number of inequalities to solve: 23456789
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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 â€” To Solve an Equation Reducible to Quadratic Form

Step 1 Write the equation in quadratic form.

Step 2 Use an appropriate â€œuâ€ 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 â€œuâ€ 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.