Mathematics: Algebraic Reasoning Study Guide for the TSIA2
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Working with Equations and Expressions
An algebraic expression looks like an equation but there is no equal sign. It is essentially a recipe describing what to do with a variable to get the desired result. For example, the expression \(5x - 3\) means to multiply the value of x by \(5\) and then subtract \(3\).
Manipulating vs. Solving – A lot of what is done in algebra falls under the heading of manipulation.
- Equations can be manipulated using the properties of equality with the goal of solving them.
- Expressions can be manipulated in different ways with the goal of putting them in different forms.
Note that expressions are never solved. For example, we can solve \(3x=27\) and say \(x=9\), but it wouldn’t make sense to look at \(4y+20\) and say that somehow, we know what value \(y\) has.
Quadratic Equations and Expressions
Quadratic equations are in the form \(ax^{2} + bx + c = 0\). This is an example of a quadratic equation:
\[3x^2 - x + 5 = 0\]with \(a = 3\), \(b = -1\), and \(c = 5\).
Manipulating
Often, a quadratic equation will be given directly in the form \(ax^2 + bx + c = 0\), such as \(x^2+4x+3=0\). Other times, certain manipulations may need to be done to work the equation into the right form.
For example \((x-2)^2=-9(x-1)\) would have to be manipulated as shown below:
\[(x-2)^2=-9(x-1)\] \[x^2-4x+4=-9x+9\] \[x^2 + 5x -5 = 0\]Solving
Quadratic equations can be solved by factoring or by using the quadratic formula.
Factoring
Factoring often involves using the pattern \((x + A)(x + B) = x^{2} + (A + B)x + AB\).
If you can get the equation to the form \((x + A)(x + B) = 0\), then \((x + A)\) or \((x + B)\) (or both) must be zero, since the product is zero.
Using the Quadratic Formula
One nice thing about the quadratic formula is that it works for any quadratic equation, factorable or not. Here is an example:
Solve \(x^2+5x+3 = 0\)
Since this is already in the form \(ax^2 + bx + c = 0\), we can see that:
\(a=1\)
\(b=5\)
\(c=3\)
Substitute the given values into the quadratic formula:
\[x = \dfrac{-2 \pm \sqrt{5^2-4 \cdot 1 \cdot 3}}{2 \cdot 1}\] \[x = \dfrac{-2 \pm \sqrt{25-12}}{2}\] \[x = \dfrac{-2 \pm \sqrt{13}}{2}\] \[x = \dfrac{-2 + \sqrt{13}}{2} \text{ or }x = \dfrac{-2 - \sqrt{13}}{2}\]Polynomials
A monomial is a variable expression of the form \(ax^{n}\).
\(a\) represents a number and is called the coefficient.
The variable is raised to the power \(n\). The value of \(n\) is often \(3\) or higher.
A polynomial is an expression of different monomials added to, or subtracted from, each other.
The highest power of the variable is the degree of the polynomial.
Each monomial member of the polynomial is called a term.
Manipulating
Often you will need to simplify a polynomial. The kinds of manipulations that are useful for this are expanding expressions using PEMDAS (doing operations in the order of parenthesis, exponents, multiplication/division, addition/subtraction), collecting like terms, factoring, and ordering from highest to lowest degree of a variable. These same manipulations, plus the usual properties of equality, are used when you need to solve polynomial equations .
Simplifying
An indication that a polynomial needs to be simplified is seeing multiple terms with the same exponent. Similar terms need to be collected.
\[3x^3+2x -4 +8x -x^3 \text{ is not simplified, }\] \[\text{but }2x^3+10x-4 \text{ is simplified}\]Another indication is seeing operations with parentheses.
\[5(2x^2-3) -2(5x^2+2)-8 \text{ is not simplified, }\] \[10x^2-15-10x^2-4+8\] \[\text{but } -11 \text{ is simplified}\]Example 1:
Consider the following unsimplified polynomial:
\[3x^2 + (x - 2)(x + 2) - 2x^2\]To simplify, follow the order of operations:
\[3x^2 + (x - 2)(x + 2) - 2x^2\] \[3x^2 + (x^2 + 2x - 2x - 4) - 2x^2\] \[3x^2 +x^2 -2x^2 +2x-2x-4\]which further simplifies to:
\[2x^2 - 4\]Example 2
Solve \((x-1)(x^2+3x+2)=0\)
Factor and get \((x-1)(x+2)(x+1)=0\)
Set each factor equal to zero and solve each for \(x\).
\[\begin{array}{lll} x-1=0 & \quad x+2=0 & \quad x+1 = 0 \\ x=1 & \quad x=-2 & \quad x=-1\\ \end {array}\]Factoring
Factoring a polynomial is the process of finding simpler polynomials that, when multiplied, will give the original polynomial as a result. Factoring may involve finding a common factor in each of the coefficients, and common factors of the variable(s). For example:
\(3x^{2} + 6x^{3}\) can be factored to \(3x^{2}(1 + 2x)\)
Often, factoring involves re-writing polynomials to match known patterns. One such pattern is:
\[(x^{2} - y^{2}) = (x + y)(x - y)\]Exponential Equations and Expressions
Guidelines for working with exponential expressions and solving exponential equations are listed below.
Working with Exponents and Roots
The expression \(a^{n}\) indicates that the number a is raised to the nth power. Here, a is the base and n is the exponent. Positive exponents indicate how many times to multiply the base times itself. \(a^{3}\) means \(a\; \times\; a\; \times\; a\).
A root is the inverse operation of raising to a positive exponent. \(\sqrt[3]{a}\) means to find a number that, when raised to the third power, equals a.
A root is equivalent to raising to a fractional exponent: \(\sqrt[3]{a} = a^{\frac{1}{3}}\)
\(\sqrt{a}\), with no root number shown, means the square root, a root number of 2.
There are several rules for working with exponents:
\[a^{1} = a\] \[a^{0} = 1\] \[1^{n} = 1\] \[a^{m}a^{n} = a^{(m+n)}\] \[\frac{a^{m}}{a^{n}} = a^{(m-n)}\] \[(a^{m})^{n} = a^{m \times n}\] \[(ab)^{m} = a^{m}b^{m}\] \[(\frac{a}{b})^{m} = \frac{a^{m}}{b^{m}}\]Manipulating
The rules for working with exponents listed above can be used to help simplify terms containing exponents or to help solve exponential equations.
Example 1:
Simplify: \(\left(\dfrac{2x^2}{3}\right)^2 \cdot \dfrac{3y}{x^4}\)
Square the first term and get:
\[\dfrac{4x^4}{9} \cdot \dfrac{3y}{x^4}\]Rearranging numerals and variables,
\[\dfrac{4}{9} \cdot \dfrac{3}{1} \cdot \dfrac{x^4}{x^4} \cdot \dfrac{y}{1}\]Simplify and get:
\[\dfrac{12}{9} \cdot 1 \cdot y\] \[\dfrac{4y}{3}\]Example 2:
Simplify \(\left(\dfrac{5}{r^2}\right)^{-1} \cdot \left(\dfrac{r}{5}\right)^{-2}\)
Change positive exponents to negative exponents by taking the reciprocal of each side, and get:
\[\left(\dfrac{r^2}{5}\right)^{1} \cdot \left(\dfrac{5}{r}\right)^{2}\]Square the right hand term, drop the exponent \(1\) and get:
\[\left(\dfrac{r^2}{5}\right) \cdot \left(\dfrac{25}{r^2}\right)\]Rearranging numerals and variables,
\[\left(\dfrac{25}{5}\right) \cdot \left(\dfrac{r^2}{r^2}\right)\]Simplify and get:
\[5\]Cancelling can be a way to do this in fewer steps.
Solving Exponential Equations
An exponential equation is one that has a variable for an exponent, so if you see an \(x\) or any other variable as an exponent in an equation, you are dealing with an exponential equation. For example, \(3^{x+1} = 27\).
Manipulating Exponential Equations
To solve an exponential equation, express each side of the equation as a power of the same base. Then set the exponents equal to each other and solve.
Example 1:
\[\text{Solve } 3^{x+1} = 27\]Rewrite \(27\) as a power of \(3\):
\[3^{x+1} = 3^3\]Set the exponents equal to each other and solve for \(x\):
\[x+1=3\] \[x=2\]Example 2:
\[\text{Solve } 4^{x+3} = 16^{x+2}\]Rewrite \(16\) as \(4^2\):
\[4^{x+3} = (4^2)^{x+2}\] \[4^{x+3} = 4^{2x+4}\]Set the exponents equal to each other and solve for \(x\):
\[x+3=2x+4\] \[x=-1\]The most common manipulation used in working with rational equations and expressions is factoring. This often leads to canceling a factor from the numerator and denominator.
Example:
Simplify \(\dfrac{2x^2+5x-3}{2x^2+7x+3}\)
Factor top and bottom and get:
\[\dfrac{(x+3)(2x-1)}{(x+3)(2x+1)}\]Cancel \(x+3\) from the top and bottom and get:
\[\dfrac{(2x-1)}{(2x+1)}\]Solving
Often, it is easier to solve them by factoring both polynomials in the hope that there is a common factor that can be canceled. Otherwise, they are solved as for any other equation by performing identical operations to both sides and simplifying until the solution is found.
Example:
\(\frac{3x - 1}{4} = \frac{x + 3}{9}\), which is solved by cross multiplying and isolating x:
\(9(3x - 1) = 4(x + 3)\), which becomes
\(27x - 9 = 4x + 12\), which becomes
\(23x = 21\), or \(x = \frac{21}{23}\).
Radical Equations and Expressions
Radical expressions are simply those that contain roots. They may include numerals and/or variables.
\[\begin{array}{llll} \text{Examples:} & \quad \sqrt{8} \times \sqrt{2} & \quad \sqrt{32x^2y^4} & \quad \dfrac{\sqrt{3x^2}} {\sqrt{9z^6}}\\ \end{array}\]Radical equations are those that contain variables within radical signs.
\[\begin{array}{llll} \text{Examples:} & \quad \sqrt{x+2} = x & \quad 3 + 3\sqrt{x}=8 & \quad \sqrt{x-2} + x = 4\\ \end{array}\]Manipulating Radical Expressions
Here are the rules that apply to simplifying radical expressions.
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Multiplication: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\)
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Division: \(\dfrac{\sqrt{a}}{\sqrt{b}} = \sqrt{\dfrac{a}{b}}\)
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Addition and subtraction: Radicals can be added or subtracted only if they have the same radicand (number under the root symbol).
Example 1:
\[\text{Simplify: } \sqrt{2}(\sqrt{8} + \sqrt{10})\] \[\text{Distribute } \sqrt{2} \text{ across }(\sqrt{8} + \sqrt{10})\] \[\sqrt{2} \cdot \sqrt{8} + \sqrt{2} \cdot \sqrt{10})\] \[\sqrt{16} + \sqrt{20})\] \[\sqrt{16} + \sqrt{4} \cdot \sqrt{5})\] \[4 + 2\sqrt{5}\]Example 2:
Add \(5\sqrt{12} + 2\sqrt{48}\)
Simplify both radicals to get the same radicand:
\[5\sqrt{12} + 2\sqrt{48}\] \[5\sqrt{4} \cdot \sqrt{3}+ 2\sqrt{16} \cdot \sqrt{3}\] \[5 \cdot 2 \cdot \sqrt{3} + 2 \cdot 4 \cdot \sqrt{3}\] \[10\sqrt{3} + 8\sqrt{3}\] \[18\sqrt{3}\]Solving
To solve a radical equation, follow these steps:
\[\begin{array}{ll} 1 & \text{Isolate the radical. (or one of the radicals if there are two)}\\ 2 & \text{Raise both sides to a power. (2 for square roots, 3 for cube roots, etc.)} \\ 3 & \text{If there is still a radical left, repeat steps 1 and 2.}\\ 4 & \text{Solve the resulting equation.}\\ 5& \text{Check the resulting solution(s). }\\ \end{array}\]Example 1:
Solve: \(\sqrt{3x-3}=3\)
Square both sides:
\[3x-3=9\]Solve:
\[3x=12\] \[x=4\]Check:
\[\sqrt{3x-3}=3\] \[\sqrt{3 \cdot 4 -3} = 3\] \[\sqrt{9} = 3\] \[3=3\]True, so 3 is a solution.
Example 2:
Solve: \(x = 3+2\sqrt{x}\)
Isolate the radical:
\[2\sqrt{x} = x-3\]Square both sides:
\[4x=x^2-6x+9\] \[x^2-10x+9=0\] \[(x-1)(x-9) = 0\]\(x = 1\) \(x=9\)
Check \(x=1\):
\[x = 3+2\sqrt{x}\] \[1 = 3+2\sqrt{1}\] \[1 = 3 + 2 \cdot 1\] \[1 = 3 + 2\]Not true, so 1 is not a solution.
Check \(x=9\):
\[9 = 3+2\sqrt{9}\] \[9 = 3 + 2 \cdot 3\] \[9 = 3 + 6\]True, so 9 is a solution.
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