Solve algebraically the simultaneous equations: x^2 + 3x + 10 = y and; 4 - 2x = y. Give answers as co-ordinates.
Where we are given two equations that equal the same thing we can substitute one into the other e.g. (1) x2 + 3x + 10 = y (2) 4 - 2x = y. Sub (2) into (1) - replace y in (1) for (4 - 2x): x2 + 3x + 10 = 4 - 2x We have a quadratic because the highest power of x is 2: (x2). To solve a quadratic we must rearrange the equation so that it is in the form: ax2 + bx + c = 0: x2 + 3x + 10 = 4 - 2x; x2 + 5x + 10 = 4 (added 2x to both sides); x2 + 5x + 6 = 0 (subtracted 4 from both sides). Now there are several ways to find x from this equation. First try factorising - if this doesn't work we can use 'complete the square' or the quadratic formula. When we factorise a quadratic we end up with something that looks like this:(x + a)(x + b) = 0. Imagine we were to expand out this equation -> Using the FOIL method of expanding, we would end up with:x2 + bx + ax + ab = 0; x2 + (a + b)x + ab = 0 (factorised ax + bx). If we compare this to our equation that we are trying to solve - we know that: a + b = 5 and; ab = 6 (we don't need to do the proof every time but remember that you must find two numbers that multiply to give the third term in the quadratic and add together to give the second term (the coefficient of x) --> this only works where the coefficient of x2 is 1. So we can first look at the factors of 6 to find which numbers multiply to give 6: 2, 3; 1, 6; -2, -3; -1, -6. Then see which add to give 5: 2 + 3 = 5; 1 + 6 = 7; -2 + -3 = -5; -1 + -6 = -7. a and b are 2 and 3 so we can say x2 + 5x + 6 = 0 is:(x + 2)(x + 3) = 0. Where we have two numbers multiplying to give 0, one of them must be 0. Since we don't know which one, we solve for the outcome of both of them equalling 0 and end up with two values of x. x + 2 = 0; x = -2. x + 3 = 0; x = -3. To find the y, we sub each x value into the easiest of the two simultaneous equations to solve (usually with the lowest power of x) - so in this case, sub into (2): 4 - 2 (-2) = y; 4 + 4 = y; y = 8 Coordinates: (-2, 8). 4 - 2(-3) = y; 4 + 6 = y; y = 10 Coordinates: (-3, 10)
