Affected Quadratic Equations.

1. The sum of the squares of two consecutive numbers is $41$. Find the numbers.

lint Let x = one number, lintand x + 1 = the other. lint Then x^2 + (x + 1)^2 = the sum of the squares; lintbut 41 = the sum of the squares.

$$\begin{gathered} \therefore x^2+(x+1{)}^2\&=41. \\ {x}^2+x^2+2x+1\&=41. \\ 2x^2+2x\&=40. \\ {x}^2+x\&=20. \end{gathered}$$

The solution of this equation gives $x=4$, or $-5$.

The positive root $4$ gives for the numbers $4$ and $5$.

The negative root $-5$ is inapplicable to the problem, as consecutive numbers are understood to be integers which follow each other in the common scale: $1$, $2$, $3$, $4\dots$.

2. In a certain nest seven times the number of birds in the nest is equal to twice the square of the number increased by $3$. Find the number.

lint Let x = the number of birds. lint Then 7x = $7$ times the number lintand 2x^2 + 3 = twice the square of the number plus $3$. As these two expressions are equal we have 2x^2 + 3 = 7x.

The solution of this equation gives $x=3$ or $x=\frac{1}{2}$.

The value $\frac{1}{2}$ is not applicable to the problem for the number of birds must be a whole number.

3. A cistern has two pipes. By one of them it can be filled $6$ hours sooner than by the other, and by both together in $4$ hours. Find the time it will take each pipe alone to fill it.

lint Let x = the number of hours it takes the smaller pipe. lint Then x - 6 = the number of hours it takes the larger pipe.

lint Therefore 1x + 1x - 6 = the part both can fill in one hour. lint But 14 = the part both can fill in one hour. 1x + 1x - 6 = 14

The solution of this equation gives $x=12$ or $x=2$.

The value $2$ is not applicable to the problem.

Therefore it takes one pipe $12$ hr. and the other $6$ hr.

4. A rug is $1$ yard longer than it is broad. The number of sq. yds. in the rug is $12$. Find its length and breadth.

lint Let x = the number of yards in the breadth. lint Then x + 1 = the number of yards in the length lintand x(x + 1) = the number of sq yds in the rug. lint Hence x(x + 1) = 12.

The solution of this equation gives $x=3$ or $x=-4$.

The dimensions are therefore $3$ yards and $4$ yards.