GCSE Revision 101 Maths Quadratics © Daniel Holloway.

Slides:



Advertisements
Similar presentations
Quadratics ax2 + bx + c.
Advertisements

SOLVING QUADRATICS General Form: Where a, b and c are constants.
QUADRATICS EQUATIONS/EXPRESSIONS CONTAINING x2 TERMS.
Solving Quadratic Equations By Keith Rachels And Asef Haider.
The Quadratic Formula for solving equations in the form
EXAMPLE 4 Choose a solution method Tell what method you would use to solve the quadratic equation. Explain your choice(s). a. 10x 2 – 7 = 0 SOLUTION a.
Surds Learning objectives Different kind of numbers
The Quadratic Formula..
Solving Quadratic Equations by Completing the Square
Factoring Polynomials
Using the Quadratic Formula to Solve a Quadratic Equation
Solving Quadratic Equations by FACTORING
Quadratics       Solve quadratic equations using multiple methods: factoring, graphing, quadratic formula, or square root principle.
Solving Quadratic Equations Section 1.3
Copyright © Cengage Learning. All rights reserved.
Section 10.5 – Page 506 Objectives Use the quadratic formula to find solutions to quadratic equations. Use the quadratic formula to find the zeros of a.
Three simple methods for solving quadratic equations
Imaginary & Complex Numbers 5-3 English Casbarro Unit 5: Polynomials.
Factoring Tutorial.
Keelen, Stevie, Nic, and Marissa, This yellow tile represents a POSITIVE one. This red tile represents a NEGATIVE one.
6.6 Quadratic Equations. Multiplying Binomials A binomial has 2 terms Examples: x + 3, 3x – 5, x 2 + 2y 2, a – 10b To multiply binomials use the FOIL.
1 Aims Introduce Quadratic Formulae. Objectives Identify how to solve quadratic equation’s using a quadratic equation.
Intermediate Tier - Algebra revision Contents : Collecting like terms Multiplying terms together Indices Expanding single brackets Expanding double.
Minds On : Factor completely: 4x 2 - 4x +1= 3x 2 +6x+9 = Determine the value of k that makes the expression a perfect square trinomial: x x +k =
2.6 Solving Quadratic Equations with Complex Roots 11/9/2012.
SOLVING QUADRATIC EQUATIONS BY COMPLETING THE SQUARE BECAUSE GRAPHING IS SOMETIMES INACCURATE, ALGEBRA CAN BE USED TO FIND EXACT SOLUTIONS. ONE OF THOSE.
Copyright © 2014, 2010, 2007 Pearson Education, Inc. 1 1 Chapter 9 Quadratic Equations and Functions.
Introduction This Chapter focuses on solving Equations and Inequalities It will also make use of the work we have done so far on Quadratic Functions and.
Quadratic Equations Learning Outcomes  Factorise by use of difference of two squares  Factorise quadratic expressions  Solve quadratic equations by.
The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.
Solving Quadratic Equations Quadratic Equations: Think of other examples?
By Kendal Agbanlog 6.1-Measurement Formulas and Monomials 6.2-Multiplying and Dividing Monomials 6.3-Adding and Subtracting Polynomials 6.4-Multiplying.
Review of Topic Equations Changing subject of formulae Inequalities.
Methods for Solving….. Factoring Factoring Square Roots Square Roots Completing the Square Completing the Square Graphing Graphing Quadratic Formula Quadratic.
Sometimes, a quadratic equation cannot be factored Example: x 2 + 7x + 3 = 0 There is not a pair of numbers that multiply to give us 3, that will also.
ALGEBRA 2 – CHAPTER 5 QUADRATICS. 5-2 PROPERTIES OF PARABOLAS.
Difference between Expression and Equation?. L.O. To be able to solve Quadratic Equations There are several ways to solve a quadratic equation Factorising.
 A method for breaking up a quadratic equation in the form ax 2 + bx + c into factors (expressions which multiply to give you the original trinomial).
Warm-Up Exercises EXAMPLE 1 Standardized Test Practice What are the solutions of 3x 2 + 5x = 8? –1 and – A 8 3 B –1 and 8 3 C 1 and – 8 3 D 1 and 8 3 SOLUTION.
11/06/ Factorizing Quadratic Equations. 11/06/ x 2 ± bx ± c x 2 ± bx ± c If this is positive, both signs are the same. The numbers ADD to.
Solving Quadratic Equations by Using the Quadratic Formula (9-5) Objective: Solve quadratic equations by using the Quadratic Formula. Use the discriminant.
Quadratic Formula Finding solutions to quadratic equations can be done in two ways : 1. Factoring – it’s a short cut. Use it if you can 2. Using the Quadratic.
College Algebra B Unit 8 Seminar Kojis J. Brown Square Root Property Completing the Square Quadratic Equation Discriminant.
Welcome! Grab a set of interactive notes
End Behavior.
GCSE Revision 101 Maths Quadratics © Daniel Holloway.
The Quadratic Formula..
Warm up.
Nature of Roots of a Quadratic Equation
Quadratic Formula Solving for X Solving for quadratic equations.
YES! The Quadratic Formula
Solving Quadratic Equations by the Quadratic Formula
Factoring Quadratic Equations
1B.1- Solving Quadratics:
6.4 Factoring and Solving Polynomial Equations
Factorising quadratics
5.4 Completing the Square.
The Quadratic Formula..
A, b and c can be any numbers
Solving Quadratic Equations by FACTORING
The Quadratic Formula..
A, b and c can be any numbers
What’s the same and what’s different?
quadratic formula. If ax2 + bx + c = 0 then
A, b and c can be any numbers
Maths Unit 25 – Solving Equations
Presentation transcript:

GCSE Revision 101 Maths Quadratics © Daniel Holloway

The Basics Factorisation of quadratics involves putting quadratic expressions back into brackets (if possible). To do this, the expression can be in the form: x2 + ax + b Where a and b are integers

Simple Factorisation We can tell how to start off the brackets by looking at the signs of the expression: x2 + ax + b = (x + ?)(x + ?) because everything is positive x2 - ax + b = (x - ?)(x - ?) because negative x negative = positive

Simple Factorisation When the second sign is a plus, both brackets are the same as the first sign x2 + ax + b = (x + ?)(x + ?) x2 - ax + b = (x - ?)(x - ?) When the second sign is a minus, both brackets are different signs x2 + ax - b = (x + ?)(x - ?) x2 - ax - b = (x + ?)(x - ?)

Simple Factorisation When a quadratic expression is in the form x2 + ax + b, we must make sure that the numbers in the brackets satisfy these rules: the b must be the product of the two numbers the a must be the sum of the numbers when the two brackets are the same sign the a must be the difference between the numbers when the two brackets are different signs

Simple Factorisation Take this example: x2 + 5x + 6 Both signs are positive, so we can begin with (x + ?)(x + ?) The numbers need to multiply to make 6 and add together to make 5 They must be 2 and 3 Since both signs are positive, the order does not matter: (x + 2)(x + 3)

Simple Factorisation Take this example: y2 – y – 90 Both signs are negative, so we can begin with (x + ?)(x - ?) The numbers need to multiply to make 90 and add together to make 1 They must be 9 and 10 Because we need to do 9 - 10 to get -1: (y + 9)(y - 10)

Test Yourself Factorise the following… a) x2 + 5x + 4 b) x2 + 7x + 10 c) y2 + 14y + 24 d) z2 + 9z + 18 e) a2 - 6a - 7 f) x2 - x - 12 g) k2 + 4k + 3 h) s2 - 18c + 32 Factorise x2 – 24x + 144

Difference of Two Squares As a rule, (a + b)(a – b) multiplies out to get a2 - b2 This type of quadratic expression where there are only two parts to it, both of which are perfect squares, separated by a minus sign, is called the difference of two squares. Examples include: a2 - 9 a2 - 25 a2 – 100 To factorise them, we simply root them and put them in the format shown above. e.g. a2 - 9 factorises to (a + 3)(a - 3)

Solving Quadratics We can solve quadratic equations in the form x2 + ax + b by first factorising it, and finding the two answers using the double brackets. e.g. Solve x2 + 6x + 5 = 0 This factorises into (x + 5)(x + 1) If x + 5 = 0, then x = -5 If x + 1 = 0, then x = -1

Factorising ax2 + bx + c We use the same method of factorisation to factorise quadratic expressions in the form ax2 + bx + c but we have to take into account the a which will be in a bracket The following slide explains how it’s done

Factorising ax2 + bx + c Factorise 3x2 + 8x + 4. The same bracket rules apply, so both brackets are positive. As 3 only has the factors 3 and 1: (3x + ?)(x + ?) Next we notice the factors of 4 are 1x4 and 2x2. The only pair here which combine with 3x1 to make 8 is 2x2: (3 x 2) + (1 x 2) = 8 So the factorised expression is: (3x + 2)(x + 2)

Solving ax2 + bx + c There are THREE ways we can solve quadratic equations in the form ax2 + bx + c as shown on the next few slides.

Via Factorisation Solve 12x2 - 28x = -15. Rearrange the equation to equal zero 12x2 - 28x + 15 = 0 Factorise the expression (2x - 3)(6x - 5) Find the two answers as before x = 1.5 and x = 25/6

Via Quadratic Formula -b ± √b2 - 4ac x = 2a Many quadratic equations cannot be factorised because they are too complicated or do not have convenient integer factors between them. This is when the quadratic formula is used: x = -b ± √b2 - 4ac 2a

Via Quadratic Formula -b ± √b2 - 4ac x = 2a Solve 5x2 - 11x – 4 = 0. Remember the quadratic formula: Substitute the values from the equation as a, b and c Solve the equation using the values, shown on the following slide x = -b ± √b2 - 4ac 2a

Via Quadratic Formula x = -b ± √b2 - 4ac 2a x = 11 ± √121 – 4(5)(-4) Solve 5x2 - 11x – 4 = 0. We know that a = 5, b = -11 and c = -4 x = -b ± √b2 - 4ac 2a x = 11 ± √121 – 4(5)(-4) 10 x = 2.52 = 11 ± √ 201 10 x = 0.32

Via Completing the Square Remember that (x + a)2 = x2 + 2ax + a2 This means that x2 + 2ax = (x + a)2 - a2 This is the principle behind completing the square, a rule used to solve quadratic equations There are two steps to completing the square, as shown on the following slide

Via Completing the Square Step 1: Rewrite x2 + 4x – 7 in the form (x + a)2 – b Note that x2 + 4x = (x + 2)2 – 4 Therefore x2 + 4x – 7 = (x + 2)2 – 4 – 7 = (x + 2)2 – 11

Via Completing the Square Step 2: Hence solve the equation x2 + 4x – 7 = 0 We have: x2 + 4x –7 = (x + 2)2 – 11 When x2 + 4x – 7 = 0, we know: (x + 2)2 – 11 = 0 (x + 2)2 = 11 Now square root both sides: x + 2 = ±√11 x = -2 ±√11 So, x = 1.32 and x = 5.32 (to 2 d.p.)