 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that, Determine the value of x that results in an equal output for both given functions and the functions are f of x is equal to 3x minus 2 and g of x is equal to x plus 3 whole square minus of 17. We know that the x coordinate of the points that results in an equal output for both functions f of x and g of x is found by the equation f of x is equal to g of x. With this key idea let us proceed to the solution. In this question we are given two functions f of x and g of x where f of x is equal to 3x minus 2 and g of x is equal to x plus 3 whole square minus of 17. We have to determine the value of x that results in equal output for both functions. Now using the key idea we know that the x coordinate of the points that results in an equal output for both functions f of x and g of x is found by the equation f of x is equal to g of x. So we have f of x is equal to g of x which implies that f of x that is 3x minus 2 is equal to g of x that is x plus 3 whole square minus of 17. Now using this result that is a plus v whole square is equal to a square plus 2 a b plus b square we will open the brackets and this implies that 3x minus 2 is equal to now x plus 3 whole square here a is equal to x and b is equal to 3 so we have x square plus 2 into x into 3 plus 3 square minus of 17 this further implies that 3x minus 2 is equal to x square plus 2 into x into 3 is 6x plus 3 square that is 9 minus of 17 and this further implies that 3x minus 2 is equal to x square plus 6x plus 9 minus 17 is minus 8. Now taking all the terms on the right hand side of the equation we get 0 is equal to x square plus 6x minus 8 minus of 3x plus 2 this further implies that 0 is equal to x square combining like terms we have plus 6x minus 3x that is plus 3x now minus 8 plus 2 is minus 6 so we have the equation x square plus 3x minus 6 is equal to 0 now we will solve this quadratic equation using quadratic formula that is x is equal to minus of b plus minus square root of b square minus of 4ac whole upon 2 into a and here in this equation the value of a is equal to 1 the value of b is equal to 3 and the value of c is equal to minus of 6 now putting these values in this formula we get x is equal to minus of 3 plus minus square root of b square that is 3 square minus 4 into a into c that is 4 into 1 into minus of 6 upon 2 into a that is 2 into 1 that is 2 which implies that x is equal to minus of 3 plus minus square root of 9 minus minus 4 into 1 is minus 4 minus 4 into minus 6 is plus 24 so we have square root of 9 plus 24 whole upon 2 which implies that x is equal to minus of 3 plus minus square root of 33 by 2 this implies that x is equal to minus 3 plus minus square root of 33 by 2 thus we say that minus of 3 plus square root of 33 whole upon 2 and minus of 3 minus square root of 33 whole upon 2 are the two values of x for which output for both the functions is equal which is the required answer this completes our session hope you enjoyed this session