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

Q: Why does the quadratic term cancel out?

When you expand $ (30 + 4x)(x + 5) $, you get $ 4x^2 + 50x + 150 $. Since the right side is also $ 4x^2 + 650 $, the $ 4x^2 $ terms cancel, leaving a simple linear equation!

Q: How do I know which expression represents what?

Daily pages: 30 + 4xNumber of days: x + 5Total pages: 4x² + 650Remember: Total = Daily × Days

Q: Can I solve this without expanding?

No, you must expand $ (30 + 4x)(x + 5) $ to see that it simplifies to a linear equation. Trying to solve it as a quadratic would be much more complicated and unnecessary!

Q: How do I verify my final answer of 15 days?

Check: Daily pages with x = 10 is $ 30 + 4(10) = 70 $ pages. Over 15 days: $ 70 × 15 = 1050 $ pages. Also: $ 4(10)^2 + 650 = 400 + 650 = 1050 $ ✓

Quadratic Equations with Linear Equation Reduction

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?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Step-by-step solution

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.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Setup: Total pages equals daily pages times number of days
Technique: Expand (30 + 4x)(x + 5) = 4x² + 50x + 150
Check: Substitute x = 10: (30 + 40)(15) = 70 × 15 = 1050 = 4(100) + 650 ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand the left side completely
Don't just multiply 30 by (x + 5) and ignore the 4x term = missing half the expansion! This gives 30x + 150 = 4x² + 650, leading to a quadratic instead of linear equation. Always distribute both terms: 30(x + 5) + 4x(x + 5).

Practice Quiz

Test your knowledge with interactive questions

Solve for x:

$$ 2(4-x)=8 $$

FAQ

Everything you need to know about this question

Why does the quadratic term cancel out?

When you expand $(30 + 4x)(x + 5)$ , you get $4x^2 + 50x + 150$ . Since the right side is also $4x^2 + 650$ , the $4x^2$ terms cancel, leaving a simple linear equation!

How do I know which expression represents what?

Daily pages: 30 + 4x
Number of days: x + 5
Total pages: 4x² + 650
Remember: Total = Daily × Days

What if I get a negative answer for x?

Check your arithmetic! In this problem, x represents a variable that leads to positive days (x + 5), so x should be positive. Negative values might mean an error in expansion or simplification.

Can I solve this without expanding?

No, you must expand $(30 + 4x)(x + 5)$ to see that it simplifies to a linear equation. Trying to solve it as a quadratic would be much more complicated and unnecessary!

How do I verify my final answer of 15 days?

Check: Daily pages with x = 10 is $30 + 4(10) = 70$ pages. Over 15 days: $70 × 15 = 1050$ pages. Also: $4(10)^2 + 650 = 400 + 650 = 1050$ ✓

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