📘 Quiz

Solution: Step 1: Recall that for a quadratic equation ax^2 + bx + c = 0 with roots r and s, the sum of roots r + s = -b/a and the product of roots rs = c/a. Step 2: Given sin(A) and cos(A) are the roots, we have sin(A) + cos(A) = -b/a and sin(A)*cos(A) = c/a. Step 3: Squaring the sum of roots: (sin(A) + cos(A))^2 = sin^2(A) + 2*sin(A)*cos(A) + cos^2(A) = b^2/a^2. Step 4: Since sin^2(A) + cos^2(A) = 1, we have 1 + 2*(c/a) = b^2/a^2. Step 5: Multiplying through by a^2 gives a^2 + 2ac = b^2. Step 6: Adding c^2 to both sides yields a^2 + 2ac + c^2 = b^2 + c^2, which simplifies to (a + c)^2 = b^2 + c^2.