Algebra Chapter 1 IB

Solving linear equations Solve the following

Solving linear equations Solve the following

Solving linear inequalities Solve the following inequality : 4(3x+1)-3(x+2)<3x+1

Quadratic equations The perimeter of a rectangle is 34 cm. Given that the diagonal is of length 13cm ad that the width is x cm, derive the equation x2-17x+60 =0. Hence find the dimensions of the rectangle.

Question 6 the garden

Question 7 A metal sleeve

Question 8 Strand of wire

Quadratic formula Use the method of completing the square to solve ax2+bx+c=0

Completing the Square Express 3x2+15x-20 in the form a(x+p)2 +q. Hence solve this equation giving your answer to 1 d.p.

The Discriminat Project  What is it?

Research and Investigate Research and investigate what the discriminant is and how do we use it to help solve and graph quadratic equations. Include your findings on a power point and email to jwathall@mail.sis.edu.hk Work in pairs!

Looking at the discriminant Using the quadratic formula solve the following equations: 1) –x2+4x-5 = 0 2) 2x2-12x+18 = 0 3) x2-5x+4 = 0 Notice what is underneath the square root sign. The first one is negative. What does this mean about the roots? The second one equals zero-what does this mean? The last one is positive so has two answers

Discriminant Notice what is underneath the square root sign This is called the Discriminant. We use this symbol to denote this . There are three possible values of this and we will look at these values and their sketches. Draw a sketch of a quadratic if the 1)  = 0 this means that there will only be one real root 2)  > 0 this means there will be two real roots 3)  < 0 you cannot take the square of a negative number yet so there are no real roots here

Discriminant and their graphs 1)  = 0 this means that there will only be one real root 2)  > 0 this means there will be two real roots 3)  < 0 you cannot take the square of a negative number yet so there are no real roots here

Example 1 Use the discriminant to determine which of these quadratic equations has two distinct real roots, equal roots or no real roots.

Another Discriminant example Example 1 Find the value of k for which x2 +kx +9 =0 has equal roots.

Discriminant example Here is a quiz on roots from As guru

Disguised quadratic equations Is this a quadratic equation? X4+5x2-14 = 0 Let’s turn it into one so that we can solve this more easily.

Sketching quadratics When sketching a quadratics these are the key features you should include: 1) Concave up or down a>0 for concave up and a<0 for concave down 2)the x and y intercepts by letting y = 0 and x = 0 3) check the discriminant for the number of roots

Example 1 Sketch y = x2 - 5x + 4 1) this is concave up as a>0 2) y intercept when x=0 so y = 4 Now Now factorise and let y= 0 Y = (x-1)(x-4) (x-1)(x-4)= 0 so x= 1, 4 3) checking the discriminant b2-4ac = 25- 4(1)(4) = positive number so two roots

The quadratic function Graphing techniques CTS 1) Looking at the vertex by completing the square 2) To find the axis of symmetry (middle of the curve) we use x= - b/2a We can sketch a quadratic if we have it in this form f(x) = a(x-h)2+k by using completing the square. This expression tells us the parabola has shifted h units to the RIGHT and k units UP. This means that the vertex (0,0) shifts to (h,k)

Example Look at Autograph for y = a(x-b)2 + c Example 1 Use the method of completing the square to sketch the following graphs 1)Y = x2+2x+3 2) Y= x2-3x-4 3) Y= 3x2 – 6x+ 4

The working and sketches 1)Y = x2+2x+3 Y = x2+2x+ 1 + 3 – 1 Y = (x+1)2+ 2 Look at Autograph (vertex (-1,2) 2) Y= x2-3x-4 Y = x2- 3x+(3/2)2 - 4 – (3/2)2 Y= (x - 3/2) 2 – 25/4 So vertex is at (3/2, -25/4) 3) Y= 3x2 – 6x+ 4 Y = 3( x2 – 2x) + 4 Y = 3(x2 – 2x +1) +4 – 3 Y= 3(x-1)2 + 1 Here vertex is (1,1) and curve is narrower by 3 Check your answers with Autograph

The sketches

Maxima & minima problems A farmer has 40 m of fencing with which to enclose a rectangular pen. Given the pen is x m wide, A) show that its area is (20x-x2) m2 B) deduce the maximum area that he can enclose

Algebraic fractions Jennifer & Vanessa Or here multiply both sides by denominator!

Ex 1k q12

Ex 1K q13