Question

The product of two consecutive odd integers is 899. Find the integers.Note: Each set of brackets represents one solution.

Answer 1

The product of two consecutive odd integers is 899.

Let x be the first odd integer.

Then (x+2) will be the second odd integer.

Their product must be equal to 899, so we can write

$x\cdot(x+2)=899$

Simplify the equation

$\begin{gathered} x^2+2x=899 \\ x^2+2x-899=0 \end{gathered}$

This is a quadratic equation that can be solved by either factoring or using the quadratic formula.

Let's use the quadratic formula.

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

For the given case, the coefficients are

a = 1

b = 2

c = -899

$\begin{gathered} x=\frac{-2\pm\sqrt[]{2^2-4(1)(-899)}}{2(1)} \\ x=\frac{-2\pm\sqrt[]{3600}}{2} \\ x=\frac{-2\pm60}{2} \\ x=\frac{-2+60}{2},\; x=\frac{-2-60}{2} \\ x=\frac{58}{2},\; x=\frac{-62}{2} \\ x=29,\; x=-31 \end{gathered}$

So, the first integer is 29

The second odd integer is x + 2 = 29 + 2 = 31

Verify the results

$29\times31=899$

Also, -31 is the first integer.

The second integer is x + 2 = -31 + 2 = -29

Verify the results

$-31\times-29=899$

Therefore, the solution is

$\mleft\lbrace29,31\mright\rbrace\; and\; \mleft\lbrace-31,-29\mright\rbrace$