1. Find in yards the side of a square whose area is 640 acres.

2. A square garden contains of an acre. Find in yards the length of its side.

3. A dealer sold a suit of clothes at as many per cent profit as the suit cost dollars. If the profit was $4, find the prime cost of the suit of clothes.

4. The width of a rectangle is of its length, and its area is 15 acres. Find its dimensions in yards.

5. Two numbers are in the ratio of 4:5, and their product is 1620. Find them.

6. Two numbers are in the ratio of 8:15, and the sum of their squares is 7225. Find them.

7. Two numbers are in the ratio of 5:13, and the difference of their squares is 5184. Find them.

8. The area of a circle = r2, π being 3.1416, and r the radius of the circle. Calculate the radius of the circle whose area is (a) 1809.6, (b) 6647.6, (c) 24,885. Calculate the

9. The surface of a sphere = 4 πr2. radius of the sphere whose surface is (a) 5026.56 square inches, (b) 45,239 square inches, (c) 101,788 square inches, (d) 123,163 square inches.

10. In the equation v2 = 2gs, v stands for the velocity

in feet per second of a cannon ball, s, the height in feet which the ball will ascend if discharged vertically upward, and g = 32 feet. If the ball ascends 5 miles, calculate in feet per second the velocity of discharge.

11. If a body is dropped from a height and falls vertically downward, the number of feet it falls in t seconds is given by the formula sgt2 (g = 32 feet). How far will a body fall in 3 seconds? How far will it fall in 8 seconds?

=

12. If a stone is dropped from the top of a tower 400 feet high, after how many seconds will it strike the earth?

CHAPTER XI

QUADRATIC EQUATIONS

123. A quadratic equation in one variable is an equation of the form ax2 + bx + c = 0, a, b, and c being known numbers or constants, a not zero.

A quadratic equation is also called an equation of the second degree.

The following are examples of quadratic equations :

x2 = 20.

x2-3x=0.

1⁄2 x2 + } x − 4 = 0.

124. A quadratic equation of the form ax2+c=0 is called an incomplete quadratic. In English text-books this form of quadratic is known as a pure quadratic.

A quadratic equation of the form ax2 + bx + c = 0, and b being different from zero, is called a complete quadratic. In English text-books a complete quadratic is generally known as an adfected quadratic. Some American texts use affected where English texts use adfected.

Example 1. Solve

5(x2 − 3x+1)− 3 (2 x2 − 5 x + 3) + 20 = 0.

SOLUTION. Removing parentheses,

5 x2-15x+5 − 6 x2 + 15 x − 9 + 20 = 0.

Combining,

-22160.

5(42 − 3 × 4+1)− 3(2 × 42 − 5 × 4 + 3) + 20 = 0.

Solve:

8.

9.

10.

11.

x2

x2

x-1

3

19.

x + 1

+

x+1 X 1

=

21.

x2

4

7

9

810

3 3

=

x+7

x2+5 x(x+7)

125. Solution of quadratics by factoring.

Example 1. Find the roots of the equation

x2+40 13 x.

SOLUTION. Transposing 13 x, x2 - 13x+40=0.

If either of these factors is equal to zero, the equation is satisfied.

If

If

x-5=0, then x=5.

x80, then x 8.

=

The roots of x2 - 13 x + 40 are 5 and 8.

Check. 52+ 40 = 13 x 5.

82+40= 13 x 8.

To solve a quadratic equation by factoring

1. Bring all the terms to the first member of the equation. 2. Factor.

3. Make each factor equal to zero.*

4. Solve the resulting simple equations.

126. Since a quadratic expression can be resolved into two factors of the first degree, a quadratic equation has two and only two roots.