Solving Equations Analytically
How can I solve equations analytically where the unknown appears only once?
- These equations can be solved by rearranging
- For one-to-one functions you can just apply the inverse
- Addition and subtraction are inverses
- Multiplication and division are inverses
- Taking the reciprocal is a self-inverse
- Odd powers and roots are inverses
- Exponentials and logarithms are inverses
- Addition and subtraction are inverses
- For many-to-one functions you will need to use your knowledge of the functions to find the other solutions
- Even powers lead to positive and negative solutions
- Modulus functions lead to positive and negative solutions
- Trigonometric functions lead to infinite solutions using their symmetries
- Even powers lead to positive and negative solutions
- Take care when you apply many-to-one functions to both sides of an equation as this can create additional solutions which are incorrect
- For example: squaring both sides
- has one solution
- has two solutions and
- For example: squaring both sides
- Always check your solutions by substituting back into the original equation
How can I solve equations analytically where the unknown appears more than once?
- Sometimes it is possible to simplify expressions to make the unknown appear only once
- Collect all terms involving x on one side and try to simplify into one term
- For exponents use
- For logarithms use
- For exponents use
How can I solve equations analytically when the equation can't be simplified?
- Sometimes it is not possible to simplify equations
- Most of these equations cannot be solved analytically
- A special case that can be solved is where the equation can be transformed into a quadratic using a substitution
- These will have three terms and involve the same type of function
- Identify the suitable substitution by considering which function is a square of another
- For example: the following can be transformed into
- using
- using
- using
- using
- using
- using
- using
- For example: the following can be transformed into
- To solve:
- Make the substitution
- Solve the quadratic equation to get y1 & y2
- Solve and
- Note that some equations might have zero or several solutions
Can I divide both sides of an equation by an expression?
- When dividing by an expression you must consider whether the expression could be zero
- Dividing by an expression that could be zero could result in you losing solutions to the original equation
- For example:
- If you divide both sides by you get which gives
- However is also a solution to the original equation
- For example:
- To ensure you do not lose solutions you can:
- Split the equation into two equations
- One where the dividing expression equals zero:
- One where the equation has been divided by the expression:
- Make the equation equal zero and factorise
- which gives
- Set each factor equal to zero and solve: and
- Split the equation into two equations
Exam Tip
- A common mistake that students make in exams is applying functions to each term rather than to each side
- For example: Starting with the equation it would be incorrect to write or
- Instead it would be correct to write and then simplify from there
Worked Example
Find the exact solutions for the following equations:
a)
.
b)
.
c)
.
Solving Equations Graphically
How can I solve equations graphically?
- To solve
- One method is to draw the graphs and
- The solutions are the x-coordinates of the points of intersection
- Another method is to draw the graph or
- The solutions are the roots (zeros) of this graph
- This method is sometimes quicker as it involves drawing only one graph
- The solutions are the roots (zeros) of this graph
- One method is to draw the graphs and
Why do I need to solve equations graphically?
- Some equations cannot be solved analytically
- Polynomials of degree higher than 4
- Equations involving different types of functions
- Polynomials of degree higher than 4
Exam Tip
- On a calculator paper you are allowed to solve equations using your GDC unless the question asks for an algebraic method
- If your answer needs to be an exact value then you might need to solve analytically to get the exact value
Worked Example
a)
Sketch the graph .
b)
Hence find the solution to .