Quadratic Equations

Excerpts

Quadratic Equations
An equation which contains the *square* of the unknown number, but no higher power, is called a **quadratic equation**.
Quadratic Equations
The square root of any number is positive or negative. Hitherto we have given only the positive value. In this chapter we shall give both values.
Quadratic Equations
This third term is the square of half the coefficient of $x$.
Quadratic Equations
Since the square root of a negative number cannot be taken, the coefficient of $x^{2}$ must be changed to $+$.
Quadratic Equations
The square root of $-5$ differs from the square root of $+5$ in that the latter can be found as accurately as we please, while the former cannot be found at all.
Quadratic Equations
The reason that every root of the equation will not always satisfy the conditions of the problem is that the problem may have certain restrictions, expressed or implied, that cannot be expressed in the equation.

Equations

Quadratic Equations
ax^{2} + bx + c = 0
Every quadratic equation can be collected into this standard form, where a, b and c are known numbers and x is the unknown.
Quadratic Equations
(x + b)^{2} = x^{2} + 2bx + b^{2}
The square of x plus b expands to x squared plus twice b times x plus b squared.
Quadratic Equations
(x - b)^{2} = x^{2} - 2bx + b^{2}
The square of x minus b expands to x squared minus twice b times x plus b squared.
Quadratic Equations
x^{2} + 2bx = c
Every affected quadratic can be brought to this form, with the x squared coefficient made 1, so that completing the square can be applied.

Problems

Exercise 71

Exercise 71, problem 1, p. 134
$5x^{2} - 2 = 3x^{2} + 6$.
Printed answer:
- $±2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-2, 2]
Exercise 71, problem 10, p. 134
$86 - 52x = 2(8 - x)(2 - 3x)$.
Printed answer:
- $±3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-3, 3]
Exercise 71, problem 11, p. 134
Find two numbers that are to each other as $3$ to $4$, and the difference of whose squares is $112$.
Printed answer:
- $12$ and $16$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 12, l: 16}
Exercise 71, problem 12, p. 134
A boy bought a number of oranges for $36$ cents. The price of an orange was to the number bought as $1$ to $4$. How many oranges did he buy, and how many cents did each orange cost?
Printed answer:
- $12$ oranges at $3$ cts.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {n: 12, p: 3}
Exercise 71, problem 13, p. 134
A certain street contains $144$ square rods, and the length is $16$ times the width. Find the width.
Printed answer:
- $3$ rods.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {w: 3}
On the STU-32 (STU, rpn):
```
144 ENTER 16 ÷
```
```
√x
```
Calculator: +3E+0; the book prints 3. Run on the calculator core at firmware 628c96c.
Exercise 71, problem 14, p. 134
Find the number of rods in the length, and in the width of a rectangular field containing $3\frac{3}{5}$ acres, if the length is $4$ times the width.
Printed answer:
- Width, $12$ rd.; length, $48$ rd.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {w: 12, L: 48}
Exercise 71, problem 2, p. 134
$3x^{2} + 1 = 2x^{2} + 10$.
Printed answer:
- $±3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-3, 3]
Exercise 71, problem 3, p. 134
$4x^{2} - 50 = x^{2} + 25$.
Printed answer:
- $±5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-5, 5]
Exercise 71, problem 4, p. 134
$(x - 6)(x + 6) = 28$.
Printed answer:
- $±8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-8, 8]
Exercise 71, problem 5, p. 134
$(x - 5)(x + 5) = 24$.
Printed answer:
- $±7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-7, 7]
Exercise 71, problem 6, p. 134
$3(x^{2} - 11) + 2(x^{2} - 5) = 82$.
Printed answer:
- $±5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-5, 5]
Exercise 71, problem 7, p. 134
$11(x^{2} + 5) + 6(3 - x^{2}) = 198$.
Printed answer:
- $±5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-5, 5]
Exercise 71, problem 8, p. 134
$5x^{2} + 3 - 2(17 - x^{2}) = 32$.
Printed answer:
- $±3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-3, 3]
Exercise 71, problem 9, p. 134
$4(x + 1) - 4(x - 1) = x^{2} - 1$.
Printed answer:
- $±3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [-3, 3]

Exercise 72

Exercise 72, problem 1, p. 137
$x^{2} - 12x + 27 = 0$.
Printed answer:
- $9$ or $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [9, 3]
Exercise 72, problem 10, p. 137
$3x^{2} - 10x + 3 = 0$.
Printed answer:
- $3$ or $\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3, Rational(1, 3)]
Exercise 72, problem 11, p. 137
$x^{2} - 14x - 51 = 0$.
Printed answer:
- $17$ or $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [17, -3]
Exercise 72, problem 12, p. 137
$34x - x^{2} - 225 = 0$.
Printed answer:
- $25$ or $9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [25, 9]
Exercise 72, problem 13, p. 137
$x^{2} + x - 20 = 0$.
Printed answer:
- $4$ or $-5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, -5]
Exercise 72, problem 14, p. 137
$x^{2} - x - 12 = 0$.
Printed answer:
- $4$ or $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, -3]
Exercise 72, problem 15, p. 137
$2x^{2} - 12x = - 10$.
Printed answer:
- $5$ or $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, 1]
Exercise 72, problem 16, p. 137
$3x^{2} + 12x - 36 = 0$.
Printed answer:
- $2$ or $-6$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, -6]
Exercise 72, problem 17, p. 137
$(2x - 1)^{2} + 9 = 6(2x - 1)$.
Printed answer:
- $2$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, 2]
Exercise 72, problem 18, p. 137
$6(9x^{2} - x) = 55(x^{2} - 1)$.
Printed answer:
- $5$ or $-11$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, -11]
Exercise 72, problem 19, p. 137
$32 - 3x^{2} - 10x = 0$.
Printed answer:
- $2$ or $-5\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, Rational(-16, 3)]
Exercise 72, problem 2, p. 137
$x^{2} - 6x + 8 = 0$.
Printed answer:
- $4$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, 2]
Exercise 72, problem 20, p. 137
$9x^{2} - 6x - 143 = 0$.
Printed answer:
- $4\frac{1}{3}$ or $-3\frac{2}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(13, 3), Rational(-11, 3)]
Exercise 72, problem 21, p. 137
$\dfrac{x}{x - 1} - \dfrac{x - 1}{x} = \dfrac{3}{2}$.
Printed answer:
- $2$ or $\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, Rational(1, 3)]
Exercise 72, problem 22, p. 137
$\dfrac{1}{x - 2} + \dfrac{2}{x + 2} = \dfrac{5}{6}$.
Printed answer:
- $4$ or $-\frac{2}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, Rational(-2, 5)]
Exercise 72, problem 23, p. 137
$\dfrac{5x + 7}{x - 1} = 3x + 11$.
Printed answer:
- $2$ or $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, -3]
Exercise 72, problem 24, p. 137
$\dfrac{7}{x + 4} - \dfrac{1}{4 - x} = \dfrac{2}{3}$.
Printed answer:
- $10$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [10, 2]
Exercise 72, problem 25, p. 137
$\dfrac{2}{x + 3} + \dfrac{x + 3}{2} = \dfrac{10}{3}$.
Printed answer:
- $3$ or $-2\frac{1}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3, Rational(-7, 3)]
Exercise 72, problem 26, p. 137
$\dfrac{2x}{x + 2} + \dfrac{x + 2}{2x} = 2$.
Printed answer:
- $2$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [2, 2]
Exercise 72, problem 27, p. 137
$\dfrac{3(x - 1)}{x + 1} - \dfrac{2(x + 1)}{x - 1} = 5$.
Printed answer:
- $\frac{1}{2}$ or $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(1, 2), -3]
Exercise 72, problem 28, p. 137
$\dfrac{2x + 5}{2x - 5} = \dfrac{7x - 5}{2x}$.
Printed answer:
- $5$ or $\frac{1}{2}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, Rational(1, 2)]
Exercise 72, problem 29, p. 137
$\dfrac{3x - 1}{4x + 7} = \dfrac{x + 1}{x + 7}$.
Printed answer:
- $7$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [7, 2]
Exercise 72, problem 3, p. 137
$x^{2} - 4 = 4x - 7$.
Printed answer:
- $3$ or $1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3, 1]
Exercise 72, problem 30, p. 137
$\dfrac{2x - 1}{x + 3} = \dfrac{x + 3}{2x - 1}$.
Printed answer:
- $4$ or $-\frac{2}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, Rational(-2, 3)]
Exercise 72, problem 31, p. 137
$\dfrac{x + 4}{x - 4} - \dfrac{x + 2}{x - 3} = 1$.
Printed answer:
- $8$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [8, 2]
Exercise 72, problem 32, p. 137
$\dfrac{4}{x - 1} - \dfrac{5}{x + 2} = \dfrac{1}{2}$.
Printed answer:
- $4$ or $-7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, -7]
Exercise 72, problem 33, p. 137
$\dfrac{2}{x - 1} = \dfrac{3}{x - 2} + \dfrac{2}{x - 4}$.
Printed answer:
- $0$ or $3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [0, 3]
Exercise 72, problem 34, p. 137
$\dfrac{5}{x - 2} - \dfrac{3}{x - 1} = \dfrac{1}{2}$.
Printed answer:
- $0$ or $7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [0, 7]
Exercise 72, problem 35, p. 137
$\dfrac{x}{7 - x} + \dfrac{7 - x}{x} = \dfrac{29}{10}$.
Printed answer:
- $5$ or $2$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [5, 2]
Exercise 72, problem 36, p. 137
$\dfrac{2x - 1}{x - 1} + \dfrac{1}{6} = \dfrac{2x - 3}{x - 2}$.
Printed answer:
- $4$ or $-1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [4, -1]
Exercise 72, problem 4, p. 137
$5x^{2} - 4x-1 = 0$.
Printed answer:
- $1$ or $-\frac{1}{5}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, Rational(-1, 5)]
Exercise 72, problem 5, p. 137
$4x - 3 = 2x - x^{2}$.
Printed answer:
- $1$ or $-3$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, -3]
Exercise 72, problem 6, p. 137
$9x^{2} - 24x + 16 = 0$.
Printed answer:
- $\frac{4}{3}$ or $\frac{4}{3}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(4, 3), Rational(4, 3)]
Exercise 72, problem 7, p. 137
$6x^{2} - 5x-1 = 0$.
Printed answer:
- $1$ or $-\frac{1}{6}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [1, Rational(-1, 6)]
Exercise 72, problem 8, p. 137
$4x + 3 = x^{2} + 2x$.
Printed answer:
- $3$ or $-1$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [3, -1]
Exercise 72, problem 9, p. 137
$16x^{2} - 16x + 3 = 0$.
Printed answer:
- $\frac{3}{4}$ or $\frac{1}{4}$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes [Rational(3, 4), Rational(1, 4)]

Exercise 73

Exercise 73, problem 1, p. 140
Find two numbers whose sum is $11$, and whose product is $30$.
Printed answer:
- $6$ and $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 6, y: 5}
Exercise 73, problem 10, p. 140
The combined ages of a father and son amount to $64$ years. Twice the father’s age exceeds the square of the son’s age by $8$ years. Find their respective ages.
Printed answer:
- Son, $10$ yr.; father, $54$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 10, f: 54}
Exercise 73, problem 2, p. 140
Find two numbers whose difference is $10$, and the sum of whose squares is $250$.
Printed answer:
- $5$ and $15$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 15, y: 5}
Exercise 73, problem 3, p. 140
A man is five times as old as his son, and the square of the son’s age diminished by the father’s age is $24$. Find their ages.
Printed answer:
- Son, $8$ yr.; father, $40$ yr.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 8, f: 40}
Exercise 73, problem 4, p. 140
A number increased by its square is equal to nine times the next higher number. Find the number.
Printed answer:
- $9$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {n: 9}
Exercise 73, problem 5, p. 140
The square of the sum of any two consecutive numbers lacks $1$ of being twice the sum of the squares of the numbers. Show that this statement is true.
Printed answer:
- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles
Exercise 73, problem 6, p. 140
The length of a rectangular court exceeds its breadth by $2$ rods. If the length and breadth were each increased by $3$ rods, the area of the court would be $80$ square rods. Find the dimensions of the court.
Printed answer:
- $5$ rd. by $7$ rd.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {l: 7, b: 5}
Exercise 73, problem 7, p. 140
The area of a certain square will be doubled, if its dimensions are increased by $6$ feet and $4$ feet respectively. Find its dimensions.
Printed answer:
- $12$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {s: 12}
On the STU-32 (STU, rpn):
```
6 ENTER 4 ×
```
```
4 ×
```
```
6 ENTER 4 + GOLD x²
```
```
+
```
```
√x
```
```
6 ENTER 4 + +
```
```
2 ÷
```
Calculator: +12E+0; the book prints 12. Run on the calculator core at firmware 628c96c.
Exercise 73, problem 8, p. 140
The perimeter of a rectangular floor is $76$ feet and the area of the floor is $360$ square feet. Find the dimensions of the floor.
Printed answer:
- $20$ ft. by $18$ ft.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {l: 20, b: 18}
Exercise 73, problem 9, p. 140
The length of a rectangular court exceeds its breadth by $2$ rods, and its area is $120$ square rods. Find the dimensions of the court.
Printed answer:
- $10$ rd. by $12$ rd.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {b: 10, l: 12}

Exercise 74

Exercise 74, problem 1, p. 141
A boat sails $30$ miles at a uniform rate. If the rate had been $1$ mile an hour less, the time of the sailing would have been $1$ hour more. Find the rate of the sailing.
Printed answer:
- $6$ miles an hour.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 6}
On the STU-32 (STU, rpn):
```
30 ENTER 4 × 1 + √x
```
```
1 + 2 ÷
```
Calculator: +6E+0; the book prints 6. Run on the calculator core at firmware 628c96c.
Exercise 74, problem 2, p. 141
A laborer built $35$ rods of stone wall. If he had built $2$ rods less each day, it would have taken him $2$ days longer. How many rods did he build a day on the average?
Printed answer:
- $7$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {x: 7}
On the STU-32 (STU, rpn):
```
2 GOLD x² ENTER 35 × 2 GOLD x² +
```
```
√x
```
```
2 + 2 ÷
```
Calculator: +7E+0; the book prints 7. Run on the calculator core at firmware 628c96c.
Exercise 74, problem 3, p. 141
A man bought flour for $$30$. Had he bought $1$ barrel more for the same sum, the flour would have cost him $$1$ less per barrel. How many barrels did he buy?
Printed answer:
- $5$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {n: 5}
On the STU-32 (STU, rpn):
```
4 ENTER 30 × 1 + √x
```
```
1 −
```
```
2 ÷
```
Calculator: +5E+0; the book prints 5. Run on the calculator core at firmware 628c96c.
Exercise 74, problem 4, p. 141
A man bought some knives for $$6$. Had he bought $2$ less for the same money, he would have paid $25$ cents more for each knife. How many knives did he buy?
Printed answer:
- $8$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {k: 8}
On the STU-32 (STU, rpn):
```
4 ENTER 6 × 2 × 1 +
```
```
√x 1 +
```
Calculator: +8E+0; the book prints 8. Run on the calculator core at firmware 628c96c.
Exercise 74, problem 5, p. 141
What number exceeds its square root by $30$?
Printed answer:
- $36$.
verified: the printed answer passed a computed check
How it was checked
- solve: passes, with the problem read into an equation {N: 36}
On the STU-32 (STU, rpn):
```
30 ENTER 4 ×
```
```
1 +
```
```
√x
```
```
1 +
```
```
2 ÷
```
```
GOLD x²
```
Calculator: +36E+0; the book prints 36. Run on the calculator core at firmware 628c96c.