Subtract 32 to both sides to the equation becomes -5x^2 + 7x + 9 = 0.
To solve this equation, we can use the quadratic formula. The quadratic formula solves equations of the form ax^2 + bx + c = 0 x = [ -b ± √(b^2 - 4ac) ] / (2a) x = [ -7 ± √(7^2 - 4(-5)(9)) ] / ( 2(-5) ) x = [ -7 ± √(49 - (-180) ) ] / ( -10 ) x = [ -7 ± √(229) ] / ( -10) x = [ -7 ± sqrt(229) ] / ( -10 ) x = 7/10 ± -sqrt(229)/10 The answers are 7/10 + sqrt(229)/10 and 7/10 - sqrt(229)/10.
domain is the valid interval for the x (input) values for the function.
the range is the valid interval for the y (result) values of the function.
so, we have a function definition and a valid domain interval for x. we therefore try the x values and see what result values they create. and that defines the range.
The square root function has no horizontal shift, so the expression under the radical is simply x with no added terms. There is only one such answer choice, which also shows the necessary translation downward by an added negative constant.