Algebra and Functions Unit 1b Quadratic Functions

Unit 1b Quadratic Functions - Objectives

Objectives By the end of the sub-unit, students should: be able to solve a quadratic equation by factorising; be able to complete the square of a quadratic expression; be able to work with quadratic functions and their graphs; know and be able to use the discriminant of a quadratic function, including the conditions for real and repeated roots; be able to solve quadratic equations, including in a function of the unknown.

Alg1 12 02 0007 Diagram Thumb Lg (binary/octet-stream)

Quadratic Equations - Introduction

A quadrat (n) is another word for a square. Biologists use a quadrat to help estimate the number of a specific plant in a field. Quadratic is an adjective meaning; describing something as square The area of a square of side x is given by the equation; $A$ = $x^2$ Hence this is known as a quadratic equation.    

220px Quadrat Sample (binary/octet-stream)

Aid585131 V4 728px Find The Roots Of A Quadratic Equation Step 11 (binary/octet-stream)

Factorising Quadratic Expressions -1

    The equation on the right involves the square of x and no higher power and is another example of a quadratic equation.   This is the same equation in factorised form.   What do you notice about the $3$ and $4$ in the factored form in relation to the $7$ and $12$ in the original form?   The sum of $3$ and $4$ is the coefficient of $4$ and the product of $3$ and $4$ gives the constant term, $12$.   To factorise a quadratic having only one lot of the square of $x$; find two numbers whose product is the coefficient of $x$ and whose sum is the constant term.   The values $-3$ and $-4$ are called the roots of the quadratic equations since the original form 'grows' from the factorised form on expanding the brackets.

Factorising Quadratic Expressions -2

  This quadratic expression has six lots of the square of $x$ The first two terms have a common factor of $3x$ and, the second two, a common factor of  $2$ This is the result of factorising the two pairs of terms.   But notice: the product of the inner, $3$ and $4$, is the same as the product of the outer, $6$ and $2$.   This is how to factorise a quadratic expression in general.

Quad Expression (binary/octet-stream)

Completing the Square

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Caption: : Geometrical explanation of completing the square

Expression such as; $$x^2$$ and $(x+b)^2$ are called perfect squares. The diagrams on the left show how a non-perfect square is written incorporating a perfect square. $$x^2+bx$$ is replaced with $(x+{\frac{b}{2}})^2$ and $({\frac{b}{2}})^2$ is subtracted.   So;          $x^2$ + $6x$  becomes $(x+{\frac{6}{2}})^2$ - $({\frac{6}{2}})^2$ = $(x+3)^2$ - $9$   Only expressions having one lot of  $x^2$  can have their square completed. So the quadratic; $ax^2$ + $bx$ must first be written as  $a(x^2$ +${\frac{b}{a}})$ which becomes, on completing the square; $a(x+{\frac{b}{a}})^2$ - ${\frac{b^2}{4a}}$ The next slide contains a video taking you threough the process step by step.

Completing the Square

Completing The Square 4 (video/mp4)

Comments by Exam Board

Teaching points Lots of practice is needed as these algebraic skills are fundamental to all subsequent work. Students must become fluent, and continue to develop thinking skills such as choosing an appropriate method, and interpreting the language in a question. Emphasise correct setting out and notation. Students will need lots of practice with negative coefficients for  the square of $x$ and be reminded to always use brackets if using a calculator. e.g. $( -2 )^2$ Include manipulation of surds when using the formula for solving quadratic equations. [Link with previous sub-unit.]  

Solving Quadratic Equations Notes

Solving Quadratic Equations (application/pdf)