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Algebra

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.

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Algebra

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  1. Algebra Chapter 1 IB

  2. Solving linear equations • Solve the following

  3. Solving linear equations • Solve the following

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

  5. 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.

  6. Question 6 the garden

  7. Question 7 A metal sleeve

  8. Question 8 Strand of wire

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

  10. 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.

  11. The Discriminat Project  What is it?

  12. 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!

  13. 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

  14. 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

  15. 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

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

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

  18. Discriminant example • Here is a quiz on roots from As guru

  19. 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.

  20. 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

  21. 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

  22. 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)

  23. 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

  24. 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

  25. The sketches

  26. 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

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

  28. Ex 1k q12

  29. Ex 1K q13

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