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 quadratic equation 2x^2 + 2x - 300 = 0 using the factorization method, we first simplify the equation. Step 1: Simplify the equation by dividing all terms by 2. (2x^2)/(2) + (2x)/(2) - (300)/(2) = (0)/(2) x^2 + x - 150 = 0 Step 2: Attempt to factor the quadratic expression x^2 + x - 150. For a quadratic equation in the form ax^2 + bx + c = 0, the factorization method involves finding two numbers that multiply to ac and add up to b. In our simplified equation x^2 + x - 150 = 0, we have a=1, b=1, and c=-150. We need to find two numbers that multiply to 1 × (-150) = -150 and add up to 1. Step 3: List pairs of integer factors of 150 and check their sum or difference. • Factors of 150: (1, 150), (2, 75), (3, 50), (5, 30), (6, 25), (10, 15). • We need two factors whose difference is 1 (since their product is negative and their sum is positive 1). • 150 - 1 = 149 • 75 - 2 = 73 • 50 - 3 = 47 • 30 - 5 = 25 • 25 - 6 = 19 • 15 - 10 = 5 Step 4: Conclude on factorization. Since none of the integer factor pairs of 150 have a difference of 1, there are no two integers that multiply to -150 and add to 1. This means the quadratic expression x^2 + x - 150 cannot be factored into two linear factors with integer coefficients. Therefore, the factorization method (using integer factors) is not suitable for solving this particular equation. The quadratic formula, as used in the previous solution, is the appropriate method for finding the exact solutions for this equation. That's 2 down. 3 left today — send the next one.