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 Depdendent Variable

 Number of equations to solve: 23456789
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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Nonlinear Equations by Substitution

Example

Solve for w: (w2 - 5w)2 - 5(w2 - 5w) = 14

 SolutionStep 1 Write the equation in quadratic form. Subtract 14 from both sides. Step 2 Use an appropriate â€œuâ€ substitution. Substitute u for w2 - 5w. Step 3 Solve the resulting equation. (w2 - 5w)2 - 5(w2 - 5w)  (w2 - 5w)2 - 5(w2 - 5w)1 - 14   u2 - 5u - 14 = 14  = 0   = 0 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. (u - 7)(u + 2) u - 7 = 0 or u + 2 u = 7 or u w2 - 5w = 7 or w2 - 5w = 0 = 0 = -2 = -2

Write each equation in standard form. Then, use the quadratic formula to solve each equation.

The equation (w2 - 5w)2 - 5(w2 - 5w) = 14 written in standard form is (w2 - 5w)2 - 5(w2 - 5w) - 14 = 0. The graph of the corresponding function, f(w) = (w2 - 5w)2 - 5(w2 - 5w) - 14 is shown.

The graph crosses the w-axis at four locations: