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Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
Humaira quadratic
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Humaira quadratic

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  • 1. Presenter:HUMAIRA FARAZ
  • 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.
  • 3. QUADRATICEQUATIONSWORDPROBLEMSQUADRATICFORMULAQUADRATICGRAPHSCOMPLETINGSQUAREMETHOD
  • 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. COMPLETINGSQUAREMETHOD
  • 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. 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. 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. = (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. = (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. = (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. 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. 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. 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. Solving Quadratic Equations byCompleting the SquareStep 4:Take thesquare rootof eachside2( 4) 36x( 4) 6x
  • 16. Solving Quadratic Equations byCompleting the SquareStep 5: Setup the twopossibilitiesand solve4 64 6 and 4 610 and 2x=xx xx
  • 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. 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. 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. Solving Quadratic Equations byCompleting the SquareStep 4:Take thesquareroot ofeach side27 47( )4 16x7 47( )4 47 474 47 474xixix
  • 21. QUADRATICFORMULA
  • 22. 242b b acxaTHE QUADRATIC FORMULA1. When you solve using completing the squareon the general formulayou get:2. This is the quadratic formula!3. Just identify a, b, and c then substitute intothe formula.20ax bx c
  • 23. 22 7 11 0x x22427 7 4(2)( 11)2(2, 7, 117 1372 Reals - Irrational42)a bb acacbSolve using the Quadratic FormulaX = 1.176 and -4.676
  • 24. SCIENTIFIC CALCULATOR

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