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Mastering Quadratic Equations: Factorization, Formula & Completing the Square

Learn how to solve quadratic equations through factorization, formula application, and completing the square method. Explore examples with detailed solutions and understand the different scenarios of solutions in a quadratic equation. Enhance your skills in math with practical application.

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Mastering Quadratic Equations: Factorization, Formula & Completing the Square

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  1. Unit 24Solving Quadratic Equations

  2. Unit 24Solving Quadratic Equations 24.1 Factorisation

  3. A quadratic equation is of the form and can be solved easily if it factorises. Example 3 Solve Solution Factorising so either or Example 4 Solve Solution Factorising so there is only one (repeated) solution Example 1 Solve Solution Factorising so either or solution or Example 2 Solve Solution Factorising so either or ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

  4. Unit 24Solving Quadratic Equations 24.2a Using the formula

  5. There is a formula to solve the quadratic equation The solution is given by Solution Here so ? ? ? Example 1 Using the formula, solve the quadratic equation Giving the solution correct to two decimal places. ? ? ? ? ? ? ? ? ? ? or ? ? or To 2 d.p. ? ?

  6. Unit 24Solving Quadratic Equations 24.2b Using the formula

  7. Again, using the formula to solve the quadratic equation: Example 2 Solve the quadratic equation Solution Here , , and ? ? ? ? ? ? ? ? ? There is only one distinct root as ? ? ?

  8. Unit 24Solving Quadratic Equations 24.2c Using the Formula

  9. Again, using the formula to solve the quadratic equation: Example 3 Solve the quadratic equation Solution Here , , and ? ? ? ? ? ? ? ? ? This has no solution as ? This is the case when ?

  10. Unit 24Solving Quadratic Equations Summary of Number of Solutions

  11. A quadratic equation can have 2,1 or 0 solutions. The following graphs illustrate these graphically and show how the number of solutions depend on the sign of which is part of the quadratic formula. No. of solutions = 0 See previous Example 3 No. of solutions = 1 See previous Example 2 ? ? No. of solutions = 2 See previous Example 1 ?

  12. Unit 24Solving Quadratic Equations 24.4 Completing the Square

  13. The formula for solving a quadratic equation comes from completing the square. Example Express in the form when a, p and q are real numbers. Solution ? ? ? So comparing coefficients, ? ? Extension What is the minimum value of ? ? ? ? ? ? This is and it occurs when ? ? Thus ?

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