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
Gradient ya curve inatolewa na (dy)/(dx) = 2x + 1. Curve inapita kwenye pointi (1, 4). a) i) Tafuta equation ya curve. Step 1: Unganisha gradient function ili kupata equation ya curve. (dy)/(dx) = 2x + 1 y = (2x + 1) dx y = (2x^2)/(2) + x + C y = x^2 + x + C Step 2: Tumia pointi (1, 4) kupata thamani ya C. Badilisha x=1 na y=4 katika equation: 4 = (1)^2 + (1) + C 4 = 1 + 1 + C 4 = 2 + C C = 4 - 2 C = 2 Step 3: Andika equation kamili ya curve. Equation ya curve ni y = x^2 + x + 2. a) ii) Tafuta thamani za x, ambapo curve inakata x-axis. Step 1: Curve inakata x-axis wakati y=0. Weka y=0 katika equation ya curve: x^2 + x + 2 = 0 Step 2: Tatua equation ya quadratic kwa x. Tumia discriminant = b^2 - 4ac. Kwa equation hii, a=1, b=1, c=2. = (1)^2 - 4(1)(2) = 1 - 8 = -7 Step 3: Tafsiri matokeo. Kwa kuwa discriminant () ni hasi ( < 0), equation haina mizizi halisi. Hii inamaanisha curve haikati x-axis. b) Bainisha eneo lililofungwa na curve na x-axis. Kutokana na matokeo ya sehemu (a) (ii), curve y = x^2 + x + 2 haikati x-axis. Kwa kuwa coefficient ya x^2 ni chanya (1 > 0), parabola inafunguka juu na iko juu ya x-axis kabisa. Kwa hivyo, hakuna eneo lililofungwa na curve na x-axis. Last one today — send me the next one 📸