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
Another one Mazel — let's solve it. Step 1: Find the y-coordinate of the point. Substitute x=2 into the equation of the curve y = 2x^2 - 3x - 6. y = 2(2)^2 - 3(2) - 6 y = 2(4) - 6 - 6 y = 8 - 6 - 6 y = 2 - 6 y = -4 The point is (2, -4). Step 2: Find the derivative of the curve to get the slope of the tangent. Differentiate y = 2x^2 - 3x - 6 with respect to x. (dy)/(dx) = (d)/(dx)(2x^2) - (d)/(dx)(3x) - (d)/(dx)(6) (dy)/(dx) = 4x - 3 Step 3: Calculate the slope of the tangent at x=2. Substitute x=2 into the derivative (dy)/(dx). m_t = 4(2) - 3 m_t = 8 - 3 m_t = 5 Step 4: Calculate the slope of the normal. The slope of the normal m_n is the negative reciprocal of the slope of the tangent m_t. m_n = -(1)/(m_t) m_n = -(1)/(5) Step 5: Find the equation of the normal using the point-slope form. Using the point (x_1, y_1) = (2, -4) and the slope m_n = -(1)/(5): y - y_1 = m_n(x - x_1) y - (-4) = -(1)/(5)(x - 2) y + 4 = -(1)/(5)(x - 2) Multiply both sides by 5 to clear the fraction: 5(y + 4) = -1(x - 2) 5y + 20 = -x + 2 Rearrange the equation to the standard form Ax + By + C = 0: x + 5y + 20 - 2 = 0 x + 5y + 18 = 0 The equation of the normal is x + 5y + 18 = 0. Send me the next one 📸