Find the roots of the equation y = 2x^2 + 5x + 2.
The root of an equation is a point where, when the equation is plotted on a graph, it crosses the x axis. The x axis is the line y = 0, so the roots can be found by setting 0 = ax2 + bx + c where a,b and c are constants. In the case of the question, the roots are the solution to 0 = 2x2 + 5x + 2. One way this can be solved is by factorisation in the form (2x + )(x + ). To find the second numbers in the brackets, first find the factors of 2, which are just 2 and 1 since it is a prime number. Then multiply one of the factors by the coefficient of x2 so that the terms add up to the coefficient of x. If we multiplied the 1 by 2 and then added the 2, we'd get 4, so we have to multiply the 2 by 2 and then add the 1, which gives the 5 we are looking for. Then the equation looks like 0 = (2x + 1)(x + 2). For the product of two numbers to equal 0, one of the numbers has to be 0. This means that either 2x+1=0, which gives x=-1/2 when rearranged, or x+2=0, which gives x=-2. Therefore the roots of the equation are x=-1/2 and x=-2. In this case, the third term was a prime number so there was only one set of factors (2 and 1), but if it had more factors then they would have to be checked, as not all of them add up to the middle term. The factorisation can be checked by expanding the brackets, (2x+1)(x+2) gives 2xx + 2x2 + 1x +12 which gives 2x2 + 5x + 2, so we know it is right. The factor of c that is multiplied by a should be in the opposite set of brackets to the one with a.
