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Write the substitution that could be used to make the equation quadratic in u. w^{1/3} -...

Question:

Write the substitution that could be used to make the equation quadratic in u.

{eq}w^{1/3} - 3w^{1/6} + 8= 0 {/eq}

Variable Substitution:

We can use the variable substitution method to transform the given equation to a quadratic form. We could use the root value of the given function and try to substitute variable letters with a different root to make it quadratic. Doing this would make factoring or division or other problem solving easier as long as we do the conversion properly.

Answer and Explanation:

Given: {eq}\displaystyle w^{1/3} - 3w^{1/6} + 8= 0 {/eq}

For this equation quadratic, we use the following for substitution through the variable u:

Let $$\begin{align} u&=w^{1/6} && \left [ \text{Start with this substitution } \right ]\\[0.2cm] \left (u \right )^{2}&=\left (w^{1/6} \right )^{2} && \left [ \text{Squaring both sides of the equation } \right ]\\[0.2cm] u^{2}&=w^{1/3} \end{align} $$

We now substitute the values of {eq}u {/eq} and {eq}u^2 {/eq} in the given equation.

$$\begin{align} \displaystyle w^{1/3} - 3w^{1/6} + 8&= 0 && \left [ \text{Given equation} \right ]\\[0.2cm] \text{Note } 1: \ u&=w^{1/6} && \left [ \text{Value of}\ u \ \text{for substitution} \right ]\\[0.2cm] \text{Note } 2: \ u^{2}&=w^{1/3} && \left [ \text{Value of}\ u^2 \ \text{for substitution} \right ]\\[0.2cm] \displaystyle u^{2} - 3u + 8&= 0 && \left [ \text{We now have a quadratic equation} \right ]\\[0.2cm] \end{align} $$

The quadratic equation in the form of {eq}u {/eq} variable is given by: {eq}\displaystyle u^{2} - 3u + 8= 0 {/eq}


Learn more about this topic:

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Solving Quadratic Equations by Substitution

from Precalculus: High School

Chapter 4 / Lesson 6
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