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Difficulty: EasyQuadratic Equations and the Quadratic Formula

A rectangular photograph has a length of xx inches and a width that is 1.51.5 inches shorter than its length. If the area of the photograph is 1010 square inches, what is the length, in inches, of the photograph?

  1. A
    2.5
  2. B
    3.25
  3. 4Answer
  4. D
    5
  5. E
    5.75

Answer

4
The correct length is 4. When the length is 4 inches, the width is 1.5 inches shorter, which is 2.5 inches. The area is the product of these dimensions: 4 * 2.5 = 10 square inches, which matches the given condition.

Step-by-Step Solution

1
Set up the equation for the area of the photograph.
x(x−1.5)=10x(x - 1.5) = 10
The area of a rectangle is found by multiplying its length (xx) by its width (x−1.5x - 1.5).
2
Distribute the xx and subtract 1010 from both sides to write the equation in standard form.
x2−1.5x−10=0x^2 - 1.5x - 10 = 0
Standard form for a quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0.
3
Multiply the entire equation by 22 to clear the decimal coefficient.
2x2−3x−20=02x^2 - 3x - 20 = 0
Working with integer coefficients makes factoring or applying the quadratic formula easier.
4
Factor the quadratic expression.
(2x+5)(x−4)=0(2x + 5)(x - 4) = 0
Factoring helps find the roots of the equation directly.
5
Solve for xx by setting each factor to zero.
x=−2.5x = -2.5 or x=4x = 4
If the product of two numbers is zero, at least one of the numbers must be zero.
6
Exclude the negative solution.
x=4x = 4
The length of a photograph must be a positive value.

Key Concept

Solving a quadratic equation by translating a word problem and factoring.
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Topics

Quadratic Equations and the Quadratic Formula

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