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
To solve the equation x = (8 - 3x)/(11 - 7x), we will first eliminate the fraction and then solve the resulting quadratic equation. Step 1: Multiply both sides of the equation by the denominator (11 - 7x). x(11 - 7x) = 8 - 3x Step 2: Expand the left side of the equation. 11x - 7x^2 = 8 - 3x Step 3: Rearrange the terms to form a standard quadratic equation ax^2 + bx + c = 0. Move all terms to one side to set the equation to zero. 0 = 7x^2 - 11x - 3x + 8 7x^2 - 14x + 8 = 0 Step 4: Solve the quadratic equation using the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a. For the equation 7x^2 - 14x + 8 = 0, we have a = 7, b = -14, and c = 8. First, calculate the discriminant = b^2 - 4ac. = (-14)^2 - 4(7)(8) = 196 - 224 = -28 Since the discriminant is negative, there are no real solutions. The solutions are complex numbers. x = -(-14) ± sqrt(-28)2(7) x = 14 ± sqrt(28)i14 We can simplify sqrt(28) as sqrt(4 × 7) = 2sqrt(7). x = 14 ± 2sqrt(7)i14 Divide both terms in the numerator by 14. x = (14)/(14) ± 2sqrt(7)i14 x = 1 ± sqrt(7)7i The solutions for x are: x = 1 + sqrt(7)7i, x = 1 - sqrt(7)7i 3 done, 2 left today. You're making progress.