Solving quadratic equations by factoringSolving quadratic equations by factoring could be many times the simplest and quickest way to solve quadratic equation as long as you know how to factor. I strongly recommend you to study or review the following important unit: Factoring Example #1: Solving quadratic equations by factoring Solve x^{2} + 3x + 2 = 0 First, you have to factor x^{2} + 3x + 2 Since the coefficient of x^{2} is 1 (x^{2} = 1x^{2}), you can factor by looking for factors of the last term (last term is 2) that add up to the coefficient of the second term (3x, coefficient is 3) 2 = 1 × 2 2 = 1 × 2 Since 1 + 2 = 3, and 3 is the coefficient of the second term, x^{2} + 3x + 2 = ( x + 2) × ( x + 1) x^{2} + 3x + 2 = 0 gives: ( x + 2) × ( x + 1) = 0 ( x + 2) × ( x + 1) = 0 when either x + 2 = 0 or x + 1 = 0 x + 2 = 0 when x = 2 x + 1 = 0 when x = 1 Let us now check x = 2 and x = 1 are indeed solutions of x^{2} + 3x + 2 = 0 (2)^{2} + 3 × 2 + 2 = 4 + 6 + 2 = 3 + 6 = 0 (1)^{2} + 3 × 1 + 2 = 1 + 3 + 2 = 3 + 3 = 0 If instead you were solving x^{2} + 3x + 2 = 0, you will do: x^{2} + 3x + 2 = 0 ( x + 2) × ( x + 1) = 0 ( x + 2) × ( x + 1) = 0 when either x + 2 = 0 or x + 1 = 0 x + 2 = 0 when x = 2 x + 1 = 0 when x = 1 Check that this are indeed the solutions Example #2: Solving quadratic equations by factoring Solve x^{2} + x 30 = 0 First, you have to factor x^{2} + x 30 30 = 30 × 1 30 = 15 × 2 30 = 6 × 5 30 = 30 × 1 30 = 15 × 2 30 = 6 × 5 Since only 6 + 5 = 1, and 1 is the coefficient of the second term(x = 1x), x^{2} + x 30 = ( x + 6) × ( x  5) x^{2} + x 30 = 0 gives: ( x + 6) × ( x  5) = 0 ( x + 6) × ( x  5) = 0 when either x + 6 = 0 or x  5 = 0 x + 6 = 0 when x = 6 x  5 = 0 when x = 5 Let us now check x = 6 and x = 5 are indeed solutions of x^{2} + x 30 = 0 (6)^{2} + 6 30 = 36  6  30 = 30  30 = 0 (5)^{2} + 5  30 = 25 + 5  30 = 30  30 = 0 If instead you were solving x^{2} + x 30 = 0, you will do: x^{2} + x  30 = 0 ( x  6) × ( x + 5) = 0 ( x  6) × ( x + 5) = 0 when either x  6 = 0 or x + 5 = 0 x  6 = 0 when x = 6 x + 5 = 0 when x = 5 Check that this are indeed the solutions Solving quadratic equations by factoring can get very tough. See below: Example #3: Solving quadratic equations by factoring 6x^{2} + 27x + 30 = 0 First factor 6x^{2} + 27x + 30 6x^{2} + 27x + 30 = ( 3x + ?) × (2x + ?) or ( 6x + ?) × (x + ?) Now, factor the last term 30 30 = 30 × 1 30 = 15 × 2 30 = 6 × 5 To get the 27x, you have to try out the cross multiplications below. There are 6 of them. Cross multiply and add! 6x x 30 1 6x + 30x = 36x 6x x 15 2 12x + 15x = 27x 6x x 6 5 30x + 6x = 36x 3x 2x 30 1 3x + 60x = 63x 3x 2x 15 2 6x + 30x = 36x 3x 2x 6 5 15x + 12x = 27x You got a couple of choices shown in bold! 6x^{2} + 27x + 30 = 0 (6x + 15) × ( x + 2)= 0 6x + 15 = 0 6x = 15 x = 15/6 x + 2 = 0 x = 2 




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