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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10. Find the minimum value of the curve 12x^2 + 24x + 13. Step 1: Identify the type of curve. The given curve is a quadratic function f(x) = 12x^2 + 24x + 13. Since the coefficient of x^2 is 12 (which is positive), the parabola opens upwards, meaning it has a minimum value at its vertex. Step 2: Find the x-coordinate of the vertex. For a quadratic function ax^2 + bx + c, the x-coordinate of the vertex is given by the formula x = -(b)/(2a). Here, a=12 and b=24. x = -(24)/(2(12)) = -(24)/(24) = -1 Step 3: Find the minimum value (y-coordinate of the vertex). Substitute the x-coordinate of the vertex (x=-1) back into the function f(x): f(-1) = 12(-1)^2 + 24(-1) + 13 f(-1) = 12(1) - 24 + 13 f(-1) = 12 - 24 + 13 f(-1) = -12 + 13 f(-1) = 1 The minimum value of the curve is 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: Express the quadratic equation in terms of its roots. If and are the roots of the quadratic equation ax^2 + bx + c = 0, then the polynomial can be factored as a(x-)(x-) = 0. Step 2: Expand the factored form. a(x-)(x-) = a(x^2 - x - x + ) = a(x^2 - (+)x + ) = ax^2 - a(+)x + a Step 3: Compare coefficients with the original equation. We have ax^2 - a(+)x + a = 0. Comparing this to the original equation ax^2 + bx + c = 0: • Comparing the coefficients of x: -a(+) = b + = -(b)/(a) • Comparing the constant terms: a = c = (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: Apply the Factor Theorem. According to the Factor Theorem, if (2x-1) is a factor of f(x), then f(c) = 0, where c is the root of 2x-1=0. Step 2: Find the root of h(x). Set h(x) = 0: 2x-1 = 0 2x = 1 x = (1)/(2) Step 3: Substitute the root into f(x) and solve for b. Since x=(1)/(2) is a root, f((1)/(2)) must be 0. f((1)/(2)) = 6((1)/(2))^2 + b((1)/(2)) + 2 = 0 6((1)/(4)) + (b)/(2) + 2 = 0 (6)/(4) + (b)/(2) + 2 = 0 (3)/(2) + (b)/(2) + 2 = 0 Multiply the entire equation by 2 to eliminate denominators: 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.