Solve the quadratic equation x^2 + x - 6 = 0
There are 3 methods for solving quadratic equations;FactorisingThe quadratic equationCompleting the SquareUsing method 1,We observe that 2 and 3 are factors of 6 and that 3 - 2 = 1 so by factoring out we have (x + 3)(x - 2) = x^2 + x - 6Now we observe that (x + 3)(x - 2) = 0 so we know that x + 3 = 0 and x - 2 = 0 are solutions to our question,Rearranging gives us that x = - 3 and x = 2We can check this by substituting each of our answers into the question, i.e. (-3)^2 + (-3) - 6 = 0 and (2)^2 + (2) - 6 = 0Using method 2,Using the quadratic equation on our question where a = 1, b = 1 and c = - 6 we have,x = (-1 plus minus squareroot( 12 - 4(1)(- 6))/(2x1)Which gives us,x = (-1 + 5)/2 and x = (-1 - 5)/2which simplifies to x = 2 and x = - 3We can check this by substituting each of our answers into the question, i.e. (-3)^2 + (-3) - 6 = 0 and (2)^2 + (2) - 6 = 0Using method 3,By completing the square we find,(x+1/2)2 - 1/4 - 6 = 0 Which rearranges to,(x + 1/2)2 = 25/4Square rooting each side we find that,x + 1/2 = plus minus 5/2Which means our answers are x = 2 and x = -3We can check this by substituting each of our answers into the question, i.e. (-3)^2 + (-3) - 6 = 0 and (2)^2 + (2) - 6 = 0In practice we wouldn't use each method as this takes too long but if you have enough time then using more than one method is a good way at checking answers.
