Teach A Level Maths Vol 1 AS Core

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

Solving Equations Module C 1 "Certain images and/or photos on this presentation are the copyrighted property of Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images"

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

Solving Equations Solving Linear Equations Ø 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.

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

Solving Equations 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 Equations 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 Equations 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.

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

Solving Equations 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 Equations 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 Equations 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.

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

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

Solving Quadratic Equations - SUMMARY Ø Zero on one side EXCEPTION: If there is no ‘x’ term write the equation as and square root. Ø 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

Solving Equations 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

Solving Equations 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.

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

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

Solving Equations Solving Quadratic Equations - SUMMARY Ø Zero on one side EXCEPTION: If there is no ‘x’ term write the equation as and square root. Ø 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

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

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

Solving Equations 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 Equations 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.

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