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

Module C1 "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

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 = )

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.

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

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

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

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.

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

[ ] 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!

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.

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.

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

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

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

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

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.

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

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.

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

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

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

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

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

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

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