


Quadratic PolynomialsCompleting the SquareWe now introduce the method of completing the square, which can be applied to solving any quadratic equation. First we deal with the case: x^{2 }+ bx + c (the leading coefficient is 1) We want to write x^{2 }+ bx + c = (x + h)^{2 }− k (known as completing the square) Note (x + a)^{2 }= x^{2 }+ 2ax + a^{2} Rearranging gives x^{2 }+ 2ax = (x + a)^{2 }− a^{2} Write b =2a or a =b/2 Hence, Example 9.Study a few more examples. ExerciseNow try to solve a few on your own. We can use this method to complete the square for any quadratic with a coeficient of 1 for x^{2}. Examplesx^{2} + 6x + 31 = (x + 3)^{2} − (3)^{2} + 31 = (x + 3)^{2} + 22 x^{2} − 11x − 3 = (x − 5.5)^{2} − (5.5)^{2} − 3 = (x − 5.5)^{2} − 33.25 ExerciseNow try some of these exercises: Completing the square allows us to find the roots of a quadratic equation. ExamplesFind the roots of x^{2 }+ 6x + 5: x^{2 }+ 6x + 5 = (x + 3)^{2 }− 3^{2} + 55 = (x + 3)^{2 }− 4 The roots are given by x^{2 }+ 6x + 5 = (x + 3)^{2 }− 4 = 0 That is, (x + 3)^{2 } = 4 Thus x + 3 = 2 or x + 3 = −2 That is, x = 2 − 3 = −1 or x = −2 − 3 = −5 Find the roots of x^{2 }+ 10x + 12 : x^{2 }+ 10x + 12 = (x + 5)^{2 }− (5)^{2} + 12 5 = (x + 5)^{2 }− 13 The roots are given by x^{2 }+ 10x + 12 = (x + 5)^{2 }− 13 = 0 That is, (x + 5)^{2 } = 13 Thus x + 5 = or x + 5 = − and so the roots are − 5 and − − 5. ExerciseNow try some of these exercises: << Roots  Quadratic Polynomials Index  The Quadratic Formula >>
