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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You're on a roll — It looks like you've sent the same equation for question 11 again. I'll provide the solution for it once more, and then I'll address the partially visible question below it. 11. Solve for x in the equation, 9^x+1 + 3^2x = 270 Step 1: Express all terms with the same base. Rewrite 9^x+1 as (3^2)^x+1. (3^2)^x+1 + 3^2x = 270 3^2(x+1) + 3^2x = 270 3^2x+2 + 3^2x = 270 Step 2: Factor out the common term 3^2x. 3^2x · 3^2 + 3^2x = 270 3^2x(3^2 + 1) = 270 3^2x(9 + 1) = 270 3^2x(10) = 270 Step 3: Isolate the exponential term. 3^2x = (270)/(10) 3^2x = 27 Step 4: Express the right side with the same base as the left side. Since 27 = 3^3: 3^2x = 3^3 Step 5: Equate the exponents and solve for x. 2x = 3 x = (3)/(2) The value of x is (3)/(2). --- For the next question, the image shows: Given y = x^2 - 2x + 6, find the... The rest of the question is cut off. Assuming the question asks for the minimum value of y or the coordinates of the vertex of the parabola: Step 1: Identify the coefficients of the quadratic equation. The equation is in the form y = ax^2 + bx + c, where a=1, b=-2, and c=6. Since a > 0, the parabola opens upwards, meaning it has a minimum value at its vertex. Step 2: Find the x-coordinate of the vertex. The x-coordinate of the vertex is given by the formula x = -(b)/(2a). x = -(-2)/(2(1)) x = (2)/(2) x = 1 Step 3: Find the minimum value of y (the y-coordinate of the vertex). Substitute the x-coordinate of the vertex (x=1) back into the equation for y. y = (1)^2 - 2(1) + 6 y = 1 - 2 + 6 y = 5 The minimum value of y is 5. The coordinates of the vertex are (1, 5). Drop the next question.