Mastering Mathematics Smartly by Wee Wen Shih

Roots of equation

One part of curve sketching question may ask students to determine the number of real roots of an equation.

The approach usually requires one to:
1. carry out some manipulations algebraically,

2. insert an additional graph,

3. determine the number of intersections between two graphs.

For example, suppose we have already sketched the graph of $y = \frac{x^2-7x+13}{x-2}$ . Now we wish to determine the number of real roots of the equation $x^4-2x^3-x^2+7x-13=0$ .

Applying step 1:
$x^3(x-2)=x^2-7x+13$
$\Rightarrow x^3=\frac{x^2-7x+13}{x-2}$ .

Applying step 2: We insert the graph of $y=x^3$ .

Applying step 3: From the sketch (go activate your GC!), both graphs intersect at two places, so the equation has two real roots.

Mastering Mathematics Smartly by Wee Wen Shih

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Roots of equation

Mastering Mathematics Smartly
by Wee Wen Shih