Question

the product of two consecutive positive numbers is 30. use x for the lower number. what is the lower number? equation?

Explanation:

Step1: Define the two numbers

Let the lower number be \( x \). Since the numbers are consecutive positive integers, the next number will be \( x + 1 \).

Step2: Set up the equation

The product of the two numbers is 30, so we can write the equation:

$$ x(x + 1) = 30 $$

Step3: Expand and simplify the equation

Expanding the left side gives:

$$ x^2 + x = 30 $$

Subtract 30 from both sides to set the equation to zero:

$$ x^2 + x - 30 = 0 $$

Step4: Factor the quadratic equation

We need to find two numbers that multiply to -30 and add to 1. Those numbers are 6 and -5. So we can factor the equation as:

$$ (x + 6)(x - 5) = 0 $$

Step5: Solve for \( x \)

Setting each factor equal to zero gives:
\( x + 6 = 0 \) or \( x - 5 = 0 \)
Solving these, we get \( x = -6 \) or \( x = 5 \). Since we are looking for positive numbers, we discard \( x = -6 \).

Answer:

The lower number is \( 5 \).