Question

the product of two consecutive integers is 420. which quadratic equation can be used to find x, the lesser number?
○ $x^2 + 1 = 420$
○ $x^2 + 2 = 420$
○ $x^2 + x = 420$
○ $x^2 + 2x = 420$

Explanation:

Step1: Define the consecutive integers

Let the lesser integer be \( x \). Then the next consecutive integer is \( x + 1 \).

Step2: Set up the product equation

The product of the two consecutive integers is \( x(x + 1) \), and this is equal to 420.

Step3: Expand the left - hand side

Using the distributive property \( a(b + c)=ab+ac \), where \( a = x \), \( b=x \) and \( c = 1 \), we have \( x(x + 1)=x^{2}+x \). So the equation is \( x^{2}+x=420 \).

Answer:

\( \boldsymbol{x^{2}+x = 420} \) (corresponding to the option \( x^{2}+x = 420 \))