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.
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Here are the solutions to problems 10, 11, and 12. 10. Find the minimum value of the curve 12x^2 + 24x + 13. The given curve is a quadratic function y = 12x^2 + 24x + 13. This is in the form y = ax^2 + bx + c, where a=12, b=24, and c=13. Since a=12 > 0, the parabola opens upwards, meaning it has a minimum value. The minimum value occurs at the vertex of the parabola, where the x-coordinate is given by x = -(b)/(2a). Step 1: Find the x-coordinate of the vertex. x = -(24)/(2(12)) x = -(24)/(24) x = -1 Step 2: Substitute this x-value back into the equation to find the minimum y-value. y = 12(-1)^2 + 24(-1) + 13 y = 12(1) - 24 + 13 y = 12 - 24 + 13 y = -12 + 13 y = 1 The minimum value of the curve is 1. 11. If the roots of ax^2 + bx + c = 0 are and , show that + = -(b)/(a) and = (c)/(a). Step 1: Use the quadratic formula to express the roots and . The roots of the quadratic equation ax^2 + bx + c = 0 are given by: = -b + sqrt(b^2 - 4ac)2a = -b - sqrt(b^2 - 4ac)2a Step 2: Find the sum of the roots, + . + = -b + sqrt(b^2 - 4ac)2a + -b - sqrt(b^2 - 4ac)2a + = (-b + sqrt(b^2 - 4ac)) + (-b - sqrt(b^2 - 4ac))2a + = -b - b + sqrt(b^2 - 4ac) - sqrt(b^2 - 4ac)2a + = (-2b)/(2a) + = -(b)/(a) Step 3: Find the product of the roots, . = (-b + sqrt(b^2 - 4ac)2a) (-b - sqrt(b^2 - 4ac)2a) This is in the form (X+Y)(X-Y) = X^2 - Y^2, where X = -b and Y = sqrt(b^2 - 4ac). = (-b)^2 - (sqrt(b^2 - 4ac))^2(2a)^2 = (b^2 - (b^2 - 4ac))/(4a^2) = (b^2 - b^2 + 4ac)/(4a^2) = (4ac)/(4a^2) = (c)/(a) Thus, we have shown that + = -(b)/(a) and = (c)/(a). 12. Find b if h(x) = 2x-1 is a factor of f(x) = 6x^2 + bx + 2. Step 1: Use the Factor Theorem. If h(x) is a factor of f(x), then f(k)=0 for the root k of h(x)=0. First, find the root of h(x) = 2x-1: 2x - 1 = 0 2x = 1 x = (1)/(2) Step 2: Substitute this root into f(x) and set f(x)=0. f((1)/(2)) = 6((1)/(2))^2 + b((1)/(2)) + 2 = 0 Step 3: Solve the equation for b. 6((1)/(4)) + (b)/(2) + 2 = 0 (6)/(4) + (b)/(2) + 2 = 0 (3)/(2) + (b)/(2) + 2 = 0 To eliminate fractions, multiply the entire equation by 2: 2((3)/(2)) + 2((b)/(2)) + 2(2) = 2(0) 3 + b + 4 = 0 b + 7 = 0 b = -7 The value of b is -7.