Question

DISCUSS UNDERSTANDING HOW TO SOLVE AN EQUATION ALGEBRAICALLY AND GRAPHICALLY.

Answer 1

Let us assume that we have to solve a polynomial equation. If the equation is linear, we can solve it very easily. Even for a quadratic equation, we can either factorize the expression or use the quadratic formula. The difficulty arises when we are required to solve an equation of degree 3 or more. At times, the rational roots theorem or the integral roots theorem, are not of much help. Further, when the roots are irrational, we face a lot of problems in solving such equations.

Let us consider a cubic equation p(x) = 2x³-x²-3x+14 = 0. Even the rational roots theorem or the integral roots theorem will require a lot of trials which will waste a lot of time. If, instead, we use a graphing calculator, we will immediately know whether the graph of p(x) crosses the X-Axis or not. If so, then p(x) has a real root at that point . A graph of p(x) is attached. We can observe that the graph of p(x) crosses the X-Axis at x = -2. Hence -2 is a root of p(x) and hence (x+2) is a factor of p(x). Now, by long division or otherwise, we can see that p(x) = 2x²(x+2)-5x(x+2)+7(x+2)= (x+2)(2x²-5x+7). Further, the graph of p(x) crosses the X-Axis at only one point. However, p(x), being a cubic, will have 3 roots. Hence, even before using the quadratic formula, we can understand that the equation2x²-5x+7= 0 will have a conjugate pair of complex roots.

A graph of p(x) is attached.