Key Concepts
- A proportion can be solved by multiplying both sides by the lowest common denominator (LCD).
- To solve a proportion by finding cross products:
- If [latex]\frac{a}{b}=\frac{c}{d}[/latex], where [latex]b \neq 0, d \neq 0[/latex], then [latex]a \cdot d = b \cdot c[/latex].
- Square root property: If [latex]x^2=k[/latex], then [latex]x = \pm \sqrt{k}[/latex]
- If [latex]k>0[/latex] the equation has two solutions
- If [latex]k=0[/latex] the equation has one solution
- If [latex]k<0[/latex] the equation has no solution
- We can remove a radical from an equation using the following two properties:
- if [latex]a=b[/latex] then [latex]a^2 = b^2[/latex]
- for [latex]x \geq 0, (\sqrt{x})^2=x[/latex]
- To solve a radical equation:
- Isolate the radical expression
- Square both sides of the equation
- Once the radical is removed, solve for the unknown
- Check your solution
Glossary
extraneous solutions: solutions that do not create a true statement when substituted back into the original equation
proportion: a equation of the form [latex]\frac{a}{b}=\frac{c}{d}[/latex], where [latex]b \neq 0, d \neq 0[/latex]
quadratic equation: can be written [latex]ax^2+bx+c=0, a \neq 0, b \ \mathrm{and} \ c[/latex] are constants
radical equation: equation containing a radical such as a square root