If the roots of a quadratic equation are alpha and beta, and we know that alpha + beta = -3 and alpha2 + beta2 = 21, find the equation whose roots are (alpha)/(beta-2) and (beta)/(alpha-2).

To find the equation whose roots are ()/(-2) and ()/(-2), we first need to find the sum and product of the original roots, and . Step 1: Find the product of the original roots, . We are given: 1. + = -3 2. ^2 + ^2 = 21 We know the identity: ( + )^2 = ^2 + ^2 + 2. Substitute the given values into the identity: (-3)^2 = 21 + 2 9 = 21 + 2 Subtract 21 from both sides: 9 - 21 = 2 -12 = 2 Divide by 2: = -6 So, the sum of the original roots is + = -3, and the product is = -6. Step 2: Calculate the sum of the new roots. Let the new roots be r_1 = ()/(-2) and r_2 = ()/(-2). The sum of the new roots is S = r_1 + r_2: S = ()/(-2) + ()/(-2) Combine the fractions by finding a common denominator: S = ((-2) + (-2))/((-2)(-2)) Expand the numerator: Numerator = ^2 - 2 + ^2 - 2 Numerator = (^2 + ^2) - 2( + ) Substitute the known values ^2 + ^2 = 21 and + = -3: Numerator = 21 - 2(-3) = 21 + 6 = 27 Expand the denominator: Denominator = - 2 - 2 + 4 Denominator = - 2( + ) + 4 Substitute the known values = -6 and + = -3: Denominator = -6 - 2(-3) + 4 = -6 + 6 + 4 = 4 Now, calculate the sum of the new roots: S = (27)/(4) Step 3: Calculate the product of the new roots. The product of the new roots is P = r_1 r_2: P = (()/(-2)) (()/(-2)) P = ()/((-2)(-2)) We already found = -6 and (-2)(-2) = 4. P = (-6)/(4) = -(3)/(2) Step 4: Form the quadratic equation. A quadratic equation with roots r_1 and r_2 is given by x^2 - Sx + P = 0. Substitute the calculated values for S and P: x^2 - ((27)/(4))x + (-(3)/(2)) = 0 x^2 - (27)/(4)x - (3)/(2) = 0 To eliminate the fractions, multiply the entire equation by the least common multiple of the denominators (which is 4): 4 ( x^2 - (27)/(4)x - (3)/(2) ) = 4(0) 4x^2 - 27x - 6 = 0 The equation is 4x^2 - 27x - 6 = 0.