VIEWS: 37 PAGES: 2 CATEGORY: Medicine POSTED ON: 8/2/2010
Creating Quadratic Equations By Using Their Roots • In our previous work, we’ve been given a quadratic equation and asked to find its roots. Using what we know about functions, we can also create a quadratic equation by using only its roots or zeros. If x = r is a root, then (x − r) is a factor of the equation. Therefore, if an equation has roots x = r1 and x = r2 , then the resulting equation is (x − r1 )(x − r2 ) = 0 . • Example: Find a quadratic equation with the following roots: 1 (a.) − and 3 (b.) 2 − 3 and 2 + 3 (c.) 1 − 2i and 1 + 2i 2 1 (a.) Since the roots are x = − and x = 3 , then the factors are 2 ⎛ 1⎞ ⎜ x + ⎟ and (x − 3) . Therefore, we have the equation: ⎝ 2⎠ ⎛ 1⎞ ⎜ x + 2 ⎟ (x − 3) = 0 ⎝ ⎠ (2x + 1)(x − 3) = 0 ← Multiply by 2 (b.) Since the roots are x = 2 − 3 and x = 2 + 3 , then the factors are (x − 2 + 3 ) and (x − 2 − 3 ) . Therefore, we have the equation: (x − 2 + 3 )(x − 2 − 3 ) = 0 x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0 x 2 − 2x − x 3 − 2x + 4 + 2 3 + x 3 − 2 3 − 3 = 0 x 2 − 2x − 2x + 4 − 3 = 0 x 2 − 4x + 1 = 0 (c.) Since the roots are x = 1 − 2i and x = 1 + 2i , then the factors are (x − 1 + 2i) and (x − 1 − 2i) . Therefore, we have the equation: (x − 1 + 2i)(x − 1 − 2i) = 0 x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4i2 = 0 x 2 − x − 2ix − x + 1 + 2i + 2ix − 2i − 4( −1) = 0 ← i2 = −1 x 2 − 2x + 5 = 0 The Sum and Product of Roots • In the previous section, we discovered how to work backwards to determine an equation using the given roots. We can also create an equation from the sum and product of its roots. • Any quadratic equation ax 2 + bx + c = 0 can be expressed in the form x 2 − (sum of the roots)x + (product of the roots) = 0 where: b sum of the roots = − a c product of the roots = a • Example: Calculate the sum and product to determine a quadratic equation with roots x = 2 ± 3 . Solution: The roots are x = 2 − 3 and x = 2 + 3 . Therefore, the sum of the roots is: (2 − 3 ) + (2 + 3 ) = 4 The product of the roots is: (2 − 3 )(2 + 3 ) = 4+2 3 −2 3 −3 =1 Therefore, the quadratic equation is x 2 − 4x + 1 = 0 .