If the minimum of x2+4x+K is -10, find the value of K. Express 4x2-3x-1 in the form a(x+b)2+c and 8+2x-x2 in the form a-(x+b)2. Find the maximum and the value of x where it occurs.

Here are the solutions to the problems. Problem 1: If the minimum of x^2 + 4x + K is -10, find the value of K. Step 1: Identify the coefficients of the quadratic equation ax^2 + bx + c. For x^2 + 4x + K, we have a=1, b=4, and c=K. Step 2: Use the formula for the minimum value of a quadratic, which is (4ac - b^2)/(4a). We are given that the minimum value is -10. -10 = (4(1)(K) - (4)^2)/(4(1)) Step 3: Solve the equation for K. -10 = (4K - 16)/(4) Multiply both sides by 4: -40 = 4K - 16 Add 16 to both sides: -40 + 16 = 4K -24 = 4K Divide by 4: K = (-24)/(4) K = -6 The value of K is -6. Problem 2: Express 4x^2 - 3x - 1 in the form a(x+b)^2 + c where a, b and c are constant. Hence find the coordinate of the turning point. Step 1: Factor out the coefficient of x^2 from the terms involving x. y = 4x^2 - 3x - 1 y = 4(x^2 - (3)/(4)x) - 1 Step 2: Complete the square inside the parenthesis. Take half of the coefficient of x ((-3)/(4)), square it, and add and subtract it. Half of (-3)/(4) is (-3)/(8). Squaring it gives ((-3)/(8))^2 = (9)/(64). y = 4(x^2 - (3)/(4)x + (9)/(64) - (9)/(64)) - 1 Step 3: Group the perfect square trinomial and distribute the 4. y = 4((x - (3)/(8))^2 - (9)/(64)) - 1 y = 4(x - (3)/(8))^2 - 4((9)/(64)) - 1 y = 4(x - (3)/(8))^2 - (9)/(16) - 1 Step 4: Combine the constant terms. y = 4(x - (3)/(8))^2 - (9)/(16) - (16)/(16) y = 4(x - (3)/(8))^2 - (25)/(16) The expression in the form a(x+b)^2 + c is 4(x - (3)/(8))^2 - (25)/(16). Step 5: Find the coordinate of the turning point. For a quadratic in the form a(x+b)^2 + c, the turning point is at (-b, c). From our expression, b = -(3)/(8) and c = -(25)/(16). So, -b = -(-(3)/(8)) = (3)/(8). The coordinate of the turning point is ((3)/(8), -(25)/(16)). Problem 3: Express 8 + 2x - x^2 in the form a - (x+b)^2 where a and b are constant. Hence find the maximum and the exact value of x where it occurs. Step 1: Rearrange the quadratic into standard form and factor out -1. y = -x^2 + 2x + 8 y = -(x^2 - 2x - 8) Step 2: Complete the square inside the parenthesis. Take half of the coefficient of x (which is -2), square it, and add and subtract it. Half of -2 is -1. Squaring it gives (-1)^2 = 1. y = -(x^2 - 2x + 1 - 1 - 8) Step 3: Group the perfect square trinomial. y = -((x - 1)^2 - 9) Step 4: Distribute the negative sign. y = -(x - 1)^2 + 9 Rearrange to the form a - (x+b)^2: y = 9 - (x - 1)^2 The expression in the form a - (x+b)^2 is 9 - (x - 1)^2. Step 5: Find the maximum value and the value of x where it occurs. The term (x-1)^2 is always greater than or equal to 0. To maximize 9 - (x-1)^2, we need to minimize (x-1)^2. The minimum value of (x-1)^2 is 0, which occurs when x-1=0, so x=1. When (x-1)^2 = 0, the expression becomes 9 - 0 = 9. The maximum value is 9. The exact value of x where it occurs is 1.