Questions: (a) Write an equation representing the fact that the product of two consecutive even integers is 80. Use x to represent the smaller integer. (b) Solve the equation from part (a) to find the two integers.

Solution Steps

Let \( x \) represent the smaller even integer. Since the integers are consecutive even numbers, the next even integer will be \( x + 2 \).

The product of the two consecutive even integers is given as 80. Therefore, the equation is: \[ x(x + 2) = 80 \]

Expand the left side of the equation: \[ x^2 + 2x = 80 \]

Subtract 80 from both sides to set the equation to zero: \[ x^2 + 2x - 80 = 0 \]

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 2 \), and \( c = -80 \): \[ x = \frac{-2 \pm \sqrt{(2)^2 - 4(1)(-80)}}{2(1)} \] \[ x = \frac{-2 \pm \sqrt{4 + 320}}{2} \] \[ x = \frac{-2 \pm \sqrt{324}}{2} \] \[ x = \frac{-2 \pm 18}{2} \]

1. \( x = \frac{-2 + 18}{2} = \frac{16}{2} = 8 \)
2. \( x = \frac{-2 - 18}{2} = \frac{-20}{2} = -10 \)

1. If \( x = 8 \), the next even integer is \( 8 + 2 = 10 \).
2. If \( x = -10 \), the next even integer is \( -10 + 2 = -8 \).

Final Answer