Solve Quadratic Equation: Finding Study Days When Reading 30+4x Pages Daily

Andrea is preparing for a History exam.

Each day she reads $30+4x$ pages of a book and in total studies for $x+5$ days.

If Andrea reads $4x^2+650$$$ pages in total, then for how many days does she study?

To solve this problem, we'll need to correctly setup and solve an equation that relates all given quantities. Here are the steps:

• Step 1: Setup the equation.
  Since Andrea reads $30 + 4x$ pages per day and studies for $x + 5$ days, the total number of pages she reads is: $(30 + 4x)(x + 5) = 4x^2 + 650$
• Step 2: Expand and simplify the equation.
  Expanding the left-hand side, we have: $(30 + 4x)(x + 5) = 30(x + 5) + 4x(x + 5)$ $= 30x + 150 + 4x^2 + 20x$ Combine like terms: $= 4x^2 + 50x + 150$ Now, equate to the total pages: $4x^2 + 50x + 150 = 4x^2 + 650$
• Step 3: Simplify and solve for $x$.
  Subtract $4x^2$ from both sides: $50x + 150 = 650$ Subtract 150 from both sides: $50x = 500$ Divide by 50: $x = 10$ Since Andrea studies for $x + 5$ days, substitute $x = 10$: $x + 5 = 10 + 5 = 15$

Therefore, Andrea studies for $\mathbf{15}$ days.