This guide provides useful resources for a wide variety of math topics. It is targeted at students enrolled in a math course or any other Centennial course that requires math knowledge and skills.

- Welcome
- Learning Math Strategies (Online)Toggle Dropdown
- Study Skills for MathToggle Dropdown
- Simply Math
- Business MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Basic Laws
- Operations on Signed numbers
- Order of Operations
- Fractions
- Decimals
- Percents
- Ratios and Proportions
- Exponents
- Statistics
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Solving Systems of Linear Equations
- Trade and Cash Discounts
- Multiple Rates of Discount
- Payment Terms and Cash Discounts
- Markup
- Simple Interest
- Compound Interest
- Nominal and Effective Interest Rates
- Ordinary Simple Annuities
- Ordinary General Annuities

- Hospitality MathToggle Dropdown
- Place Value in Decimal Number Systems
- Arithmetic Operations
- Order of Operations
- Basic Laws
- Prime Factorisation and Least Common Multiple
- Fractions
- Decimals
- Percents
- Exponents
- Units of Measures
- Fluid Ounces and Ounces
- Metric Measures
- Yield Percent
- Recipe Size Conversion
- Ingredient Ratios
- Food-Service Industry Costs

- Engineering Math
- Basic Laws
- Order of Operations
- Prime Factorisation and Least Common Multiple
- Fractions
- Exponents
- Radicals
- Reducing Radicals
- Factoring
- Rearranging Formulas
- Solving Linear Equations
- Areas and Volumes of Figures
- Congruence and Similarity
- Functions
- Domain and Range of Functions
- Basics of Graphing
- Transformations
- Graphing Linear Functions
- Graphing Quadratic Functions
- Solving Systems of Linear Equations
- Solving Quadratic Equations
- Solving Higher Degree Equations
- Trigonometry
- Graphing Trigonometric Functions
- Graphing Circles and Ellipses
- Exponential and Logarithmic Functions
- Complex Numbers
- Number Bases in Computer Arithmetic
- Linear Algebra
- Calculus
- Set Theory
- Modular Numbers and Cryptography
- Statistics
- Problem Solving Strategies

- Upgrading / Pre-HealthToggle Dropdown
- Basic Laws
- Place Value in Decimal Number Systems
- Decimals
- Significant Digits
- Prime Factorisation and Least Common Multiple
- Fractions
- Percents
- Ratios and Proportions
- Exponents
- Radicals
- Reducing Radicals
- Metric Conversions
- Factoring
- Solving Linear Equations
- Solving Quadratic Equations
- Functions
- Domain and Range of Functions
- Polynomial Long Division
- Exponential and Logarithmic Functions
- Statistics

- Nursing MathToggle Dropdown
- Arithmetic Operations
- Order of Operations
- Place Value in Decimal Number Systems
- Decimals
- Fractions
- Percents
- Ratios and Proportions
- Nutrition Labels
- Interpreting Drug Orders
- Oral Dosages
- Dosage Based on Size of the Patient
- Parenteral Dosages
- Intravenous (IV) Administration
- Infusion Rates for Intravenous Piggyback (IVPB) Bag
- General Dosage Rounding Rules

- Transportation MathToggle Dropdown
- PhysicsToggle Dropdown

An equation is a **quadratic equation** if it can be written in the form:

\(ax^2+bx+c=0\),

where \(a,b,c\) are known values, \(a \ne 1\), and \(x\) is some unknown variable. It has degree of 2 since the quadratic polynomial has degree 2 (i.e. highest exponent of all monomials in the polynomial is 2: \(x^2\)).

Recall the methods we can use to solve quadratic equations such as factoring or using the quadratic formula (review these on the Solving Quadratic Equations page). These only work for solving quadratic equations, but what if we wanted to solve equations of higher degrees (i.e. degree 3 or higher)?

To solve higher degree equations, we can use substitution to convert the given equation into a quadratic equation, then solve the quadratic equation to determine the solutions to the original equation.

For example, suppose we have the equation:

\(ax^4+bx^2+c=0\)

If we let \(z=x^2\), then substitute this into the original equation, we can rewrite it as:

\(a(x^2)^2+b(x^2)+c=0\)

\(\Rightarrow az^2+bz+c=0\),

which is a quadratic equation that we can solve (by factoring or using the quadratic equation).

Then after solving, we can set the solutions for \(z\) equal to \(x^2\), then solve for \(x\).

**Tip:** Don't forget to find the solutions of the __original__ equation after solving the rewritten equation!

1. Solve \(x^6+5x^3+6=0\).

**Solution:**

Since we want to rewrite this equation as a quadratic equation, we use substitution by letting \(z=x^3\). So the equation becomes:

\((x^3)^2+5(x^3)+6=0\)

\(\Rightarrow z^2+5z+6=0\)

We can now solve this quadratic equation by factoring, giving us:

\((z+2)(z+3)=0\)

\(\Rightarrow z=-2\) __or__ \(z=-3\)

Finally, we solve for \(x\) using \(z=x^3\):

\(z=-2=x^3\) __or__ \(z=-3=x^3\)

\(\Rightarrow\) \(x=\sqrt[3]{-2}\) __or__ \(x=\sqrt[3]{-3}\)

We can also use this substitution method to solve other types of equations (not only ones involving polynomials), seen in this following example:

2. Solve \(\frac{x^4+x^2-20}{x^2+5}=0\).

**Solution:**

We can rewrite the rational expression on the left side by substituting with \(z=x^2\):

\(\frac{(x^2)^2+(x^2)-20}{(x^2)+5}=0\)

\(\Rightarrow \frac{z^2+z-20}{z+5}=0\)

Now we can factor the numerator to simplify the rational expression, making note of any variable restrictions (since the denominator cannot equal 0):

\(\frac{\cancel{(z+5)}(z-4)}{\cancel{(z+5)}}=0, \ z \ne 5\)

\(\Rightarrow z-4=0\)

\(\Rightarrow z=4\)

Finally, we solve for \(x\) using \(z=x^2\):

\(z=4=x^2\)

\(\Rightarrow\) \(x=2\) __or__ \(x=-2\)

__Note:__ Recall that there are two solutions to \(x^2=a\) (for any value \(a>0\), if there are no restrictions on the variable \(x\)):

\(x=+\sqrt{a}\) __or__ \(x=-\sqrt{a}\)

Designed by Matthew Cheung. This work is licensed under a Creative Commons Attribution 4.0 International License.

- Last Updated: Nov 24, 2023 12:54 PM
- URL: https://libraryguides.centennialcollege.ca/mathhelp
- Print Page

chat loading...