1. “Teach A Level Maths” Vol. 1: AS Core Modules
5: Solving Equations
© Christine Crisp

2. Module C1
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3. Expressions, Equations and Identities
e.g is an expression
The value of the expression can be found for any value of the unknown, x
e.g.
e.g is a linear equation
e.g is a quadratic equation
These equations can be solved. There is one value satisfying the 1st equation and two values which satisfy the 2nd equation.
e.g is an identity
An identity is true for all values of the unknown.
( Identities are sometimes written with instead of = )

4. Solving Linear Equations
Collect the terms containing the unknown on one side of the equation and the constants on the other
e.g.
Linear equations only have constants and x-terms without powers.

5. Two factors multiplied together = 0,
Solving Quadratic Equations
e.g. 1
Do NOT cancel x as a solution will then be lost.
Get zero on one side
Try to factorise
( Common factor )
Two factors multiplied together = 0,
so one must be zero. or

6. Solving Quadratic Equations
e.g. 2
Zero on one side
( Trinomial ) or
Try to factorise
Two factors multiplied together = 0, so one factor must equal zero. or or

7. Solving Quadratic Equations
e.g. 3 or
Multiply by -1
Trinomial
Try to factorise
Two factors multiplied together = 0, so one factor must be zero. or

8. Solving Quadratic Equations
e.g. 4
Multiply by x
In this example there is no linear term. Instead of getting 0 on the r.h.s. we can square root directly.
N.B.

9. Exercises
Solve the following quadratic equations
Solutions 1. 2. 3. 4. 5. 6.

10. [ ] A useful tip: If a quadratic equation is written as
then if is a perfect square, the quadratic will factorise
e.g. 1
The quadratic factorises!
[ ]
e.g. 2
The quadratic does not factorise!

11. Solving Quadratic Equations
e.g. 5
This quadratic doesn’t factorise so complete the square
To solve for x, we need to square root, so we isolate the squared term on the left of the equal sign (l.h.s.)
Square rooting
N.B. 2 Solutions!
These answers are exact but can be given as approximate decimals.

12. Solving Quadratic Equations
The method used in the last example can be generalised to give us a formula which is easier to use when the coefficient of is not 1
The formula will be proved but you don’t need to know the proof.
However, you must memorise the result.

13. Proof of the Quadratic Formula
Consider
Divide by a:
Complete the square:

14. Solving Quadratic Equations
e.g. 6 Solve the equation
Solution: or

15. Solving Quadratic Equations - SUMMARY
Zero on one side
EXCEPTION: If there is no ‘x’ term write the equation as and square root.
Try to factorise
Common Factors
Trinomial factors
If there are factors, factorise and solve
If there are no factors, complete the square ( if a = 1 ) or use the formula

16. Exercises
Use the most efficient method to solve the following quadratic equations: 1.
Solution:
Complete the Square 2.
Solution:
Use the formula. 3.
Solution: Factorise

18. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

19. Expressions, Equations and Identities
The value of the expression can be found for any value of the unknown, x
e.g is an expression
e.g is a quadratic equation
These equations can be solved. There is one value satisfying the 1st equation and two values which satisfy the 2nd equation.
e.g is an identity
An identity is true for all values of the unknown.
e.g.
e.g is a linear equation
( Identities can be written as but only for emphasis.)
Expressions, Equations and Identities

20. Linear equations only have constants and x-terms without powers.
Solving Linear Equations
Collect the terms containing the unknown on one side of the equation and the constants on the other
e.g.
Linear equations only have constants and x-terms without powers.

21. Zero on one side
Try to factorise
If there are no factors, complete the square ( if a = 1 ) or use the formula
Common Factors
Trinomial factors
If there are factors, factorise and solve
EXCEPTION: If there is no ‘x’ term write the equation as and square root.
Solving Quadratic Equations - SUMMARY

22. Two factors multiplied together = 0,
e.g. 1 or
Get zero on one side
( Common factor )
Two factors multiplied together = 0,
so one must be zero.
Try to factorise
Do NOT cancel x as a solution will then be lost.
Solving Quadratic Equations

23. e.g. 2
Zero on one side
( Trinomial )
Two factors multiplied together = 0, so one factor must equal zero.
Try to factorise or
Solving Quadratic Equations

24. e.g. 3
Multiply by -1
Trinomial
Two factors multiplied together = 0, so one factor must be zero.
Try to factorise or
Solving Quadratic Equations

25. e.g. 4
In this example there is no linear term. Instead of getting 0 on the r.h.s. we can square root directly.
Multiply by x
N.B.
Solving Quadratic Equations

26. e.g. 5
This quadratic doesn’t factorise so complete the square
To solve for x, we need to square root, so we isolate the squared term on the left of the equal sign (l.h.s.)
Square rooting
These answers are exact but can be given as approximate decimals.
N.B. 2 Solutions!
Solving Quadratic Equations

27. e.g. 6 Solve the equation
Solution: or
Solving Quadratic Equations