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
Welcome back Dàññý — missed you this week. To solve the quadratic equation x^2 + 6x - 7 = 0 using the completing the square method, we follow these algebraic steps: Step 1: Move the constant term to the right side of the equation. x^2 + 6x = 7 Step 2: Take half of the coefficient of the x term, square it, and add it to both sides of the equation. The coefficient of x is 6, so half of it is 3, and 3^2 = 9. x^2 + 6x + 9 = 7 + 9 Step 3: Factor the left side as a perfect square and simplify the right side. (x+3)^2 = 16 Step 4: Take the square root of both sides. Remember to include both positive and negative roots. x+3 = ±sqrt(16) x+3 = ± 4 Step 5: Solve for x by isolating it. x = -3 ± 4 This gives two solutions: x_1 = -3 + 4 = 1 x_2 = -3 - 4 = -7 Here is the simple MATLAB code that implements these steps: `matlab % MATLAB code to solve x^2 + 6x - 7 = 0 using the completing the square method % Define coefficients of the quadratic equation ax^2 + bx + c = 0 a = 1; b = 6; c = -7; % Step 1: Move the constant term to the right side and normalize by 'a' % The equation becomes x^2 + (b/a)x = -c/a % Since a=1 for this equation, it simplifies to x^2 + bx = -c b_normalized = b / a; % Coefficient of x term after dividing by a c_constant_on_right = -c / a; % Constant term on the right side after dividing by a % Step 2: Calculate the term needed to complete the square % This term is (b_normalized / 2)^2 term_to_add = (b_normalized / 2)^2; % Step 3: Add the term to both sides % The left side becomes (x + b_normalized/2)^2 % The right side becomes c_constant_on_right + term_to_add right_side_value = c_constant_on_right + term_to_add; % Step 4: Take the square root of both sides % x + b_normalized/2 = +/- sqrt(right_side_value) % Use 'complex' to handle cases where right_side_value might be negative (complex roots) if right_side_value < 0 sqrt_part = sqrt(complex(right_side_value)); else sqrt_part = sqrt(right_side_value); end % Step 5: Solve for x x1 = -b_normalized / 2 + sqrt_part; x2 = -b_normalized / 2 - sqrt_part; % Display the solutions fprintf('The quadratic equation is x^2 + 6x - 7 = 0.'); fprintf('Solutions using the completing the square method:'); fprintf('x1 = %f', x1); fprintf('x2 = %f', x2); ` When you run this MATLAB code, the output will be: The quadratic equation is x^2 + 6x - 7 = 0. Solutions using the completing the square method: x1 = 1.000000 x2 = -7.000000 The solutions are x_1 = 1, x_2 = -7. Drop the next question 📸