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.
ScanSolve AI Answer
Here are the solutions for the given equations. i) Solve the given equation by factorization: 12x^2 - 11x + 2 = 0 Step 1: Identify coefficients and find two numbers that multiply to a · c and add to b. For 12x^2 - 11x + 2 = 0, we have a=12, b=-11, c=2. We need two numbers that multiply to 12 × 2 = 24 and add up to -11. These numbers are -3 and -8. Step 2: Rewrite the middle term using these numbers. 12x^2 - 3x - 8x + 2 = 0 Step 3: Factor by grouping. 3x(4x - 1) - 2(4x - 1) = 0 Step 4: Factor out the common binomial. (4x - 1)(3x - 2) = 0 Step 5: Set each factor to zero and solve for x. 4x - 1 = 0 4x = 1 x = (1)/(4) 3x - 2 = 0 3x = 2 x = (2)/(3) The solutions are x = (1)/(4), (2)/(3). ii) Solve the given equation: 2 · 2^2x - 3 · 2^x + 1 = 0 Step 1: Rewrite the equation using exponent rules and substitute a variable. The equation is 2 · (2^x)^2 - 3 · 2^x + 1 = 0. Let y = 2^x. Substituting this into the equation gives a quadratic equation in terms of y. 2y^2 - 3y + 1 = 0 Step 2: Factor the quadratic equation for y. We need two numbers that multiply to 2 × 1 = 2 and add up to -3. These numbers are -1 and -2. 2y^2 - y - 2y + 1 = 0 y(2y - 1) - 1(2y - 1) = 0 (2y - 1)(y - 1) = 0 Step 3: Solve for y. 2y - 1 = 0 2y = 1 y = (1)/(2) y - 1 = 0 y = 1 Step 4: Substitute back y = 2^x and solve for x. For y = (1)/(2): 2^x = (1)/(2) 2^x = 2^-1 x = -1 For y = 1: 2^x = 1 2^x = 2^0 x = 0 The solutions are x = -1, 0. That's 2 down. 3 left today — send the next one.