Solving problems of the type   $ax^2 + \frac{b}{c} = \frac{d}{e}$


In this section, we continue to solve algebraic equations containing the square of our variable x, this time with some fractions involved, just to give you more practice with those and with reducing ratios of square roots. Here's an example:

$$3 x^2 - \frac{2}{3} = \frac{1}{5},$$

As usual for these, remember that

$$\sqrt{x^2} = (\sqrt{x})^2 = x$$

Notice that the squaring and square-root operations (functions, really) undo one another. Applying them in sequence is like doing nothing at all to the variable x. They allow us to "liberate" a variable from a root or square.

The first step to solving such a problem is to do the easy stuff first, namely, move the $-4$ to the right by adding it to both sides:

$$ \begin{align} &3 x^2 - \frac{2}{3} = \frac{1}{5} \\ &\underline{\phantom{000} + \frac{2}{3} + \frac{2}{3}} \\ &\phantom{0000}x^2 = \frac{13}{15} \\[5pt] &\phantom{00000}x = ± \sqrt{\frac{13}{15}} \end{align}$$

Here we can't reduce the roots, but you should always make sure to reduce them to the smallest possible representation of the original(s).


What if   $c - b < 0?$


If $c - b < 0,$ then we end up taking the square root of a negative number. That's possible with a little trick you may or may not know. Check out sections on quadratic functions and complex numbers to learn more, but here's a very short version of how to do it.

The imaginary number   $i$

Essentially, we make up a new number, $i = \sqrt{-1}.$ Now the square root of any negative number like -x can be expressed like

$$\sqrt{-x} = \sqrt{-1 \cdot x} = \sqrt{-1}\sqrt{x} = i\sqrt{x}$$

To take the square root of a negative number, simply find the root of the absolute value and append the imaginary number, $i.$


Practice

You can practice problems of this type below. They are generated randomly. Practice them until you consistently get them right; these should eventually be very easy for you. Although you can compare your calculator-generated answers, try to arrive at exact answers by reducing roots as much as possible.





Hit answer to view solution.

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Inverse operations

In this section, we'll refer often to inverse operations. Inverse operations are opposite, and one can be used to undo the action of the other.

  • Addition and subtraction are inverse operations.

  • Multiplication and division are inverse operations.
  • The square-root and squaring functions or operations are inverses. Each undoes the action of the other.

Other Algebra practice problems

There are a number of these pages you can use for algebra practice. Just pick the rough type of problem you need to work on.

Type 1:   $ax + by = c$

Type 2:   $\frac{a}{b}x + \frac{c}{d} = \frac{e}{f}$

Type 3:   $\frac{a}{x} + b = c$

Type 4:   $\frac{a}{x} + \frac{b}{c} = \frac{d}{e}$

Type 5:   $x^2 + b = c$

Type 6:   $ax^2 + \frac{b}{c} = \frac{d}{e}$

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