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# Solving Nonlinear Equations by Substitution

Example

Solve for w: w -2 - 2w -1 = 15

 SolutionStep 1 Write the equation in quadratic form. Subtract 15 from both sides.Write w -2 as (w -1)2. Step 2 Use an appropriate â€œuâ€ substitution.Substitute u for w -1. Step 3 Solve the resulting equation. Factor the new equation. 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. w -2 - 2w -1  w -2 - 2w -1 - 15 (w -1)2 - 2(w -1)1 - 15   u2 - 2u - 15   (u - 5)(u + 3) u - 5 = 0 or u + 3 u = 5 or u w -1 = 5 or w -1 = 15  = 0 = 0   = 0   = 0 = 0 = -3= -3 Use the definition of a negative exponent. = -3 Multiply by the LCD, w. 1 = 5w or 1 = -3w Divide by the coefficient of w. = w

So, there are two solutions: The equation w -2 - 2w -1 = 15 written in standard form is w -2 - 2w -1 - 15 = 0. The graph of the corresponding function, f(w) = w -2 - 2w -1 - 15  is shown. The graph crosses the w-axis at two locations, Note that w -1 can be written as Since w is in the denominator it cannot equal 0. Therefore the line w = 0 is a vertical asymptote.