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 Sunday 18th of February

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# Solving Nonlinear Equations by Substitution

Some nonlinear equations can be rewritten so that they can be solved using the methods for solving quadratic equations.

Recall the general form of a quadratic equation: ax2 + bx + c = 0.

The variable of the first term, ax2, has an exponent of 2.

The variable of the second term, bx, has an exponent of 1.

The third term, c, is a constant.

If we can rewrite an equation in quadratic form then we can solve the equation by using a method for solving a quadratic equation, such as factoring or by using the quadratic formula.

For example, consider the equation x4 + 3x2 - 10 = 0.

We can write the first term with an exponent of 2: x4 = (x2)2

We can write the second term with an exponent of 1: x2 = (x2)1

The third term is a constant.

To make it easier to see the quadratic form, we use the substitution u = x2. That is, we replace x2 with u.

 Original equation. x4 + 3x2 - 10 = 0 Think of x4 as (x2)2. (x2)2 + 3(x2)1 - 10 = 0 Substitute u for x2. u2 + 3u - 10 = 0

The last equation is in quadratic form. We can solve it by factoring or by using the quadratic formula.

After solving for u, we can use u = x2 to find the values for x.