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Completing the Square

Completing the Square. Slideshow 16, Mathematics Mr Richard Sasaki, Room 307. Objectives. Recall how to solve quadratic equations through factorisation Learn how to “complete the square” to solve equations for . Solving through factorisation.

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Completing the Square

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  1. Completing the Square Slideshow 16, Mathematics Mr Richard Sasaki, Room 307

  2. Objectives • Recall how to solve quadratic equations through factorisation • Learn how to “complete the square” to solve equations for .

  3. Solving through factorisation As we learned last class, to factorise a quadratic equation, we look for two numbers that add together to make the -coefficient and multiply to make the constant. Example Solve - 10x + 24 = 0. - 1+ 24 = 0 (-6)(-4)= 0 So = 6 or 4.

  4. Completing the Square Completing the square is similar and attempts to make the expression in show in the form (x + a)2. This is always possible for an expression + bx+ c if we move some of the constant c to the right-hand side. This can be very difficult if you forget the method!

  5. Completing the Square Solve + 6x - 16 = 0. Example + 6- 16 = 0 It makes it easier if we move the constant to the other side. + 6= 16 We need (x + 3)2 = a. (Note 3 is half of 6 as this would produce 3x + 3x. To get (x + 3)2 we half 6 and then square it. 3 = 9. We add this to each side. + 6+ = 16 + 9 9 Now write the left as (x + a)2. +3= = (+3)2 = 25 -8 = -3 Square root both sides and solve!

  6. Completing the Square You must show your working! There is a lot to remember! Instructions: 1. Move the constant to the right. 2. Half and square the middle coefficient and add this to each side. 3. Write the left part as . 4. Square root each side and solve for . Example Solve + 8x - 20 = 0. + 8= 20 Halve and square. + 8+ = 20 + 16 16 (+4)2 = 36 +4=6 = -4 6 = 2 or -10

  7. Completing the Square + 4x = 21 + 10x = -24 + 10x + 25 = 1 + 4x + 4 = 21 + 4 = 1 = 25 = 5 or -5 = 1 or -1 = 0 (zero isn’t positive or negative.)

  8. Completing the Square When completing the square it is easy to make a small mistake, be careful! When we have a negative coefficient of x, the symbol within the squared brackets will also be negative. Make sure you carry the negative symbol!

  9. Negative Cases Solve - 4x - 12 = 0. Example The addition to each side is always positive as (½x)2 > 0. - 4= 12 - 4+ 4= 12+ 4 - 2)= 16 - 2 =4 or -4 =6 or -2 Hint: The number in the bracket above is always half of the original -coefficient!

  10. Completing the Square , 16, 64, 25/4

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