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1. 1. Presenter:HUMAIRA FARAZ
2. 2. BASICALGEBRA• ax+b ; 2x-7• Difference b/wexpression andequationsSOLVE BASICEQUATIONS• 2x-8=4• 2/x + 7 = 5/xQUADRATICEQUATIONS? • ax² + bx + c = 0 (a ≠ 0)SOLVEQUADRATICEQUATIONS.• Square root property.• Algebraic Formula• Factorization.
4. 4. Factorization (Examples)Example 1x2– 2x – 24 = 0x2– 6x + 4x – 24 = 0(x + 4)(x – 6) = 0x + 4 = 0 x – 6 = 0 x = –4 x = 6Example 2x2– 8x + 11 = 0x2– 8x + 11 is prime;therefore, anothermethod must be usedto solve this equation.
5. 5. COMPLETINGSQUAREMETHOD
6. 6. (x + 4)2x + 4x+4x2 4x4x 16Completing The SquareSome quadratic functions can written as a perfect squares.x2 + 8x + 16 x2 + 10x + 25(x + 5)2x + 5x 5x25+5 5xx2We can show thisgeometrically whenthe coefficient of x ispositive.When we writeexpressions in thisform it is known ascompleting thesquare.
7. 7. Completing The SquareSome quadratic functions can written as a perfect square.x2 + 8x + 16(x + 4)2(x - 2)2x2 - 4x + 4Similarly when the coefficient of x is negative:What is the relationshipbetween the constant termand the coefficient of x?The constant term is always (half the coefficient of x)2.
8. 8. Completing The Squarex2 + 3x + 2.25(x + 1.5)2(x - 3.5)2x2 - 7x + 12.25When the coefficient of x is odd we can still write a quadratic expression as a non-perfect square, provided that the constant term is (half the coefficient of x)2
9. 9. = (x + 2)2= (x - 3)2x2 + 4xx2 - 6xCompleting The SquareThis method enables us to write equivalent expressions for quadratics ofthe form ax2 + bx. We simply half the coefficient of x to complete thesquare then remember to correct for the constant term.- 4- 9
10. 10. = (x + 1.5)2= (x - 3.5)2x2 + 3x x2 - 7xCompleting The SquareThis method enables us to write equivalent expressions for quadratics ofthe form ax2 + bx. We simply half the coefficient of x to complete thesquare then remember to correct for the constant term.- 2.25 - 12.25
11. 11. = (x + 5)2= (x + 2)2= (x - 1)2= (x - 6)2x2 + 4x + 3 x2 + 10x + 15x2 - 2x + 10 x2 - 12x - 1Completing The SquareWe can also write equivalent expressions for quadratics of the formax2 + bx + c. Again, we simply half the coefficient of x to complete the squareand remember to take extra care in correcting for the constant term.- 1 - 10+ 9 - 37
12. 12. Solving Quadratic Equationsby Completing the SquareSolve the followingequation bycompleting thesquare:Step 1: Movequadratic term, andlinear term to leftside of the equation28 20 0x x28 20x x
13. 13. Solving Quadratic Equationsby Completing the SquareStep 2: Find the termthat completes the squareon the left side of theequation. Add that term toboth sides.28 =20 +x x  21( ) 4 then square it, 4 162828 2016 16x x
14. 14. Solving Quadratic Equationsby Completing the SquareStep 3: Factorthe perfectsquare trinomialon the left sideof the equation.Simplify the rightside of theequation.28 2016 16x x
15. 15. Solving Quadratic Equations byCompleting the SquareStep 4:Take thesquare rootof eachside2( 4) 36x( 4) 6x
16. 16. Solving Quadratic Equations byCompleting the SquareStep 5: Setup the twopossibilitiesand solve4 64 6 and 4 610 and 2x=xx xx
17. 17. Completing the Square-Example #2Solve the followingequation by completingthe square:Step 1: Move quadraticterm, and linear term toleft side of the equation,the constant to the rightside of the equation.22 7 12 0x x22 7 12x x
18. 18. Solving Quadratic Equations byCompleting the SquareStep 2: Find the termthat completes the squareon the left side of theequation. Add that termto both sides.The quadratic coefficientmust be equal to 1 beforeyou complete the square, soyou must divide all termsby the quadraticcoefficient first.2222 722 2 27 1272=-12 +6x xx xxx    21 7 7 49( ) then square it,2 62 4 4 172 49 4916 1762 6x x
19. 19. Solving Quadratic Equationsby Completing the SquareStep 3: Factorthe perfectsquare trinomialon the left side ofthe equation.Simplify the rightside of theequation.2227627 96 494 16 167 47449 4916 1166x xxx
20. 20. Solving Quadratic Equations byCompleting the SquareStep 4:Take thesquareroot ofeach side27 47( )4 16x7 47( )4 47 474 47 474xixix