Questions: A student opens a mathematics book to two facing pages. The product of the page numbers is 420. Find the page numbers.

Solution Steps

To find the two facing page numbers whose product is 420, we can set up a system of equations. Let \( x \) be the first page number and \( x+1 \) be the second page number. The product of these two numbers is given by \( x(x+1) = 420 \). We can solve this quadratic equation to find the value of \( x \).

Let the first page number be \( x \). Since the pages are consecutive, the second page number will be \( x + 1 \). The product of the two page numbers is given by the equation: \[ x(x + 1) = 420 \]

Rearranging the equation gives us a standard quadratic form: \[ x^2 + x - 420 = 0 \]

To solve the quadratic equation, we first calculate the discriminant \( D \): \[ D = b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-420) = 1 + 1680 = 1681 \]

Using the quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \), we find the roots: \[ x = \frac{-1 \pm \sqrt{1681}}{2 \cdot 1} \] Calculating the square root: \[ \sqrt{1681} = 41 \] Thus, the roots are: \[ x_1 = \frac{-1 + 41}{2} = 20 \quad \text{and} \quad x_2 = \frac{-1 - 41}{2} = -21 \]

Since page numbers must be positive integers, we take \( x = 20 \). Therefore, the first page number is \( 20 \) and the second page number is: \[ 20 + 1 = 21 \]

Final Answer