 Hello students, let's work out the following problem. It says if x is equal to 2 by 3 and x is equal to Minus 3 are the roots of the equation ax square plus 7x plus b is equal to 0 find the values of a and b So let's now move on to the solution the given equation is a x square plus 7x plus b is Equal to 0. Let us name this as 1 Now we are given that x is equal to 2 by 3 is the root of 1 Therefore x is equal to 2 by 3 satisfies this equation So we put x is equal to 2 by 3 in this equation. So we have a into 2 by 3 square plus 7 into 2 by 3 plus b is equal to 0 So we have a into 4 by 9 plus 14 by 3 plus b is equal to 0. Now taking LCM, LCM would be 9 and simplifying we have 4a plus 14 into 3 is 42 Plus 9b is equal to 0. So this implies 4a plus 9b is Equal to minus 42. Let us name this as a Now we are given x is equal to minus 3 is the root of 1 so we have a into minus 3 square plus 7 into minus 3 plus b is equal to 0 so we have 9a minus 21 plus b is equal to 0. So this implies 9a Plus b is equal to 21. Let us name this as equation b. Now we'll solve equation a and b for small a and b. Now we'll solve equation a and b by equating the coefficient of a and b. So here we'll equate the coefficient of b and for that we multiply equation b by 9 and then we'll subtract equation b from so we have Equation a as 4a plus 9b is equal to minus 42 and we have to multiply equation b by 9. So we have 81a plus 49b and 21 into 9 is 189 Now we have to subtract and Since we are subtracting sign will change So 4a minus 81a is minus 77a Plus 9a gets cancelled with minus 9b minus 42 minus 189 is minus 231 so this implies a is equal to minus 231 upon minus 77 and that implies a is equal to 3. Now put a is equal to 3 in any one of the two equations either equation a or b So we'll put a is equal to 3 in a So we have 4a a is 3 plus 9b is equal to minus 42. So this implies 12 plus 9b is equal to minus 42 and this implies 9b is equal to minus 42 minus 12 and this implies 9b is equal to minus 54 and this implies v is equal to minus 54 by 9 So this implies b is equal to minus 6 hence the value of a is 3 and v is minus 6 So this completes the question and the session. Bye for now take care. Have a good day