Quadratic Quadratic equationsequations

A reviewA review

Factorising Quadratics to Factorising Quadratics to solve!- four methodssolve!- four methods

1) Common factors you must take out any 1) Common factors you must take out any common factors first xcommon factors first x22+19x=0+19x=0

x(x+19) = 0 x= 0,-19x(x+19) = 0 x= 0,-19 2) Recognition these are called cookie cutters 2) Recognition these are called cookie cutters

(a+b)(a+b)22, (a-b), (a-b)22 or or

(a+b)(a-b)=0(a+b)(a-b)=0

Proof to perfect squareProof to perfect square Proof to difference of two squaresProof to difference of two squares 3) Cross method3) Cross method 4) Quadratic formula4) Quadratic formula

A warm up activity- solve A warm up activity- solve the following the following

1) x1) x22+6x+5=0+6x+5=0

2) x2) x22-3x-40=0-3x-40=0

3) X3) X22-9=0-9=0

4) x4) x22-11x=0-11x=0

5) x5) x22-169=0-169=0

6) 9x6) 9x22-25=0-25=0

Homework Homework

1616thth January January Ex 23, 24, Quadratic Formula 25 Ex 23, 24, Quadratic Formula 25 Choose all or odd questions Choose all or odd questions

Cross Method- Cross Method- Factorising QuadraticsFactorising Quadratics

Solve xSolve x2 2 +15x+56 = 0+15x+56 = 0 There are three steps to follow:There are three steps to follow: Step 1 draw a cross and write the factors of 5mStep 1 draw a cross and write the factors of 5m22

Step 2 write down the factors of the constant 56 so that Step 2 write down the factors of the constant 56 so that cross ways they add up to the middle term which is 15x. cross ways they add up to the middle term which is 15x. Remember here the sign of the constant is very important. Remember here the sign of the constant is very important. Negative means they are different and positive means the Negative means they are different and positive means the signs are the samesigns are the same

Step 3 write from left to right top to bottom the factorised Step 3 write from left to right top to bottom the factorised form.form.

Another example using Another example using cross methodcross method

Solve : xSolve : x22-3x-40 = 0-3x-40 = 0

The minus 40 tells me the factors The minus 40 tells me the factors have different signs.have different signs.

Yet another example of Yet another example of cross methodcross method

Solve xSolve x22+3x-180=0+3x-180=0

How does the cookie How does the cookie cutter work?cutter work?

(x+2)(x+2)22 = x = x22 + 4x + 4 + 4x + 4 You should recognise that the right You should recognise that the right

hand side is a perfect square- a hand side is a perfect square- a cookie cutter resultcookie cutter result

There are three cookie cutter resultsThere are three cookie cutter results What are they?What are they?

Perfect SquarePerfect Square

Look at this: what is (a+b)Look at this: what is (a+b)22 =? =? a ba b

aa

bb

There are many ways to There are many ways to solve quadratic solve quadratic

equationsequations Factorise any common factors first! Factorise any common factors first! A) Cross methodA) Cross method B) Standard results cookie cutters B) Standard results cookie cutters Now we are going to look at:Now we are going to look at:

C) solving quadratics by using theC) solving quadratics by using the quadratic formula!quadratic formula!

The Quadratic formulaThe Quadratic formula

Remember this:Remember this:

The Quadratic formulaThe Quadratic formula

Ok let’s prove this using the method Ok let’s prove this using the method of completing the square.of completing the square.

An animation deriving thisAn animation deriving this

Some examples hereSome examples here

The Quadratic Formula The Quadratic Formula

Using the quadratic formula. Using the quadratic formula. Sometimes you cannot use the cross Sometimes you cannot use the cross method because the solutions of the method because the solutions of the quadratic is not a whole number!quadratic is not a whole number!

Example solve the following giving Example solve the following giving you solution correct to 3 sig figyou solution correct to 3 sig fig

3x3x22-8x+2 = 0-8x+2 = 0

Solving quadratic Solving quadratic equations equations

Example 1 Solve xExample 1 Solve x22 + 3x – 4 = 0 + 3x – 4 = 0

Example 2 Solve 2xExample 2 Solve 2x22 – 4x – 3 = 0 – 4x – 3 = 0 This doesn’t work with the methods we This doesn’t work with the methods we

know so we use a formula to help us solve know so we use a formula to help us solve this.this.

Quadratic formula Quadratic formula

Form purple math an introForm purple math an intro

A songA song

Where does it come from? Where does it come from?

Example Example

Example Solve 2xExample Solve 2x22 – 4x – 3 = 0 – 4x – 3 = 0 a = 2, b = -4 and c = -3a = 2, b = -4 and c = -3

Using you brain!Using you brain!

Only use the quadratic formula to solve Only use the quadratic formula to solve an equation when you cannot factorise an equation when you cannot factorise it by usingit by using

A) cookie cutterA) cookie cutter B) cross methodB) cross method

Some word problemsSome word problems

The height h m of a rocket above the The height h m of a rocket above the ground after t seconds is given by ground after t seconds is given by

h =35t -5th =35t -5t22. When is the rocket 50 m . When is the rocket 50 m above the ground?above the ground?