This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Here are the solutions to the problems: Problem 1: Solve the equation x^4 + 2x^3 - 7x^2 - 8x + 12 = 0 Step 1: Find integer roots using the Rational Root Theorem. Let P(x) = x^4 + 2x^3 - 7x^2 - 8x + 12. We test integer divisors of the constant term 12 (± 1, ± 2, ± 3, ± 4, ± 6, ± 12). For x=1: P(1) = (1)^4 + 2(1)^3 - 7(1)^2 - 8(1) + 12 = 1 + 2 - 7 - 8 + 12 = 0. So, x=1 is a root, and (x-1) is a factor. Step 2: Perform polynomial division (or synthetic division) by (x-1). Dividing P(x) by (x-1) gives x^3 + 3x^2 - 4x - 12. So, the equation becomes (x-1)(x^3 + 3x^2 - 4x - 12) = 0. Step 3: Find roots for the cubic factor Q(x) = x^3 + 3x^2 - 4x - 12. Again, test integer divisors of the constant term 12. For x=2: Q(2) = (2)^3 + 3(2)^2 - 4(2) - 12 = 8 + 3(4) - 8 - 12 = 8 + 12 - 8 - 12 = 0. So, x=2 is a root, and (x-2) is a factor of Q(x). Step 4: Perform polynomial division (or synthetic division) by (x-2) on Q(x). Dividing Q(x) by (x-2) gives x^2 + 5x + 6. So, the equation becomes (x-1)(x-2)(x^2 + 5x + 6) = 0. Step 5: Factor the quadratic term x^2 + 5x + 6. This quadratic factors as (x+2)(x+3). So, the equation is (x-1)(x-2)(x+2)(x+3) = 0. Step 6: Set each factor to zero to find the roots. x-1 = 0 x = 1 x-2 = 0 x = 2 x+2 = 0 x = -2 x+3 = 0 x = -3 The solutions are: x = 1, 2, -2, -3 --- Problem 2: Geometry Problem Given: • Triangle ABC is a right-angled triangle at B. • Length of side AB = 5 cm. • Length of side BC = 12 cm. • Point D is on AC such that BD is perpendicular to AC. Required: Find the length of BD. Step 1: Find the length of the hypotenuse AC using the Pythagorean theorem in ABC. Since ABC is a right-angled triangle at B, we have: AC^2 = AB^2 + BC^2 AC^2 = (5)^2 + (12)^2 AC^2 = 25 + 144 AC^2 = 169 AC = sqrt(169) AC = 13 cm Step 2: Use the area of ABC. The area of a triangle can be calculated using the formula (1)/(2) × base × height. We can calculate the area of ABC in two ways: 1. Using AB as base and BC as height (since B = 90^): Area( ABC) = (1)/(2) × AB × BC Area( ABC) = (1)/(2) × 5 cm × 12 cm Area( ABC) = (1)/(2) × 60 cm^2 Area( ABC) = 30 cm^2 2. Using AC as base and BD as height (since BD is perpendicular to AC): Area( ABC) = (1)/(2) × AC × BD Step 3: Equate the two expressions for the area to find BD. (1)/(2) × AC × BD = 30 cm^2 Substitute the value of AC: (1)/(2) × 13 cm × BD = 30 cm^2 13 × BD = 60 cm BD = (60)/(13) cm The length of BD is: BD = (60)/(13) cm