Question 1

One positive number is 4 larger than another number. The sum of their squares is 106. Find the numbers.
A) 1, 5
B) 8, 11
C) 5, 9
D) 6, 10

Accepted Answer

Let x be the smaller number, then the larger number is x + 4. From the given information, we have the equation x^2 + (x+4)^2 = 106. Expanding and simplifying, we get 2x^2 + 8x - 90 = 0. Factoring out 2, we get x^2 + 4x - 45 = 0. The factors of -45 that differ by 4 are -5 and 9, so x = -9 or x = 5. Since we are looking for a positive number, we choose x = 5, which gives the larger number as x + 4 = 9. Therefore, the numbers are 5 and 9, which is choice C.

Question 2

Find three negative consecutive integers such that the product of the two smaller integers is 142 more than ten times the largest integer.
A) -9, -8, -7
B) -7, -6, -5
C) -12, -11, -10
D) -10, -9, -8
E) -11, -10, -9

Accepted Answer

Let the three consecutive integers be x, x-1, and x-2.Then, (x-1)(x-2) = 142 + 10xx^2 - 3x + 2 = 142 + 10xx^2 - 13x - 140 = 0(x-14)(x+10) = 0x = 14 or x = -10If x = 14, then the three integers are 14, 13, and 12.If x = -10, then the three integers are -10, -11, and -12.Therefore, the correct answer is A.

Question 3

Suppose that the width of a certain rectangle is 4 inches less than its length. The area is numerically 11 less than twice the perimeter. Find the length and width of the rectangle.
A) 7 in., 11 in.
B) 5 in., 9 in.
C) 4 in., 8 in.
D) 10 in., 14 in.

Accepted Answer

We can use algebra to solve the problem. Let x be the length of the rectangle. Then the width is x-4. The area of the rectangle is x(x-4) = x^2 - 4x. The perimeter of the rectangle is 2(x + x-4) = 4x-8. Twice the perimeter is 8x-16. We are told that the area is numerically 11 less than twice the perimeter, so we can write the equation:
x^2 - 4x = 2(4x-8) - 11
Simplifying this equation gives:
x^2 - 4x = 8x - 27
Bringing all the terms to one side gives:
x^2 - 12x + 27 = 0
This quadratic equation factors as:
(x-3)(x-9) = 0
So x = 3 or x = 9. We reject x=3 since the width would be negative. Therefore, the length of the rectangle is 9 inches and the width is 5 inches (which is 4 inches less than the length). So the answer is (B) 5 in., 9 in.

Question 4

Suppose that one leg of a right triangle is 7 feet shorter than the other leg. The hypotenuse is 2 feet longer than the longer leg. Find the lengths of all three sides of the right triangle.
A) 6 feet, 16 feet, 18 feet
B) 6 feet, 17 feet, 18 feet
C) 10 feet, 17 feet, 19 feet
D) 8 feet, 15 feet, 17 feet
E) 9 feet, 16 feet, 18 feet

Accepted Answer

Let the shorter leg be x feet. Then, the longer leg is x + 7 feet, and the hypotenuse is x + 7 + 2 = x + 9 feet. According to the Pythagorean theorem, $x^2 + (x + 7)^2 = (x + 9)^2$. Solving this equation gives x = 8, so the sides are 8 feet, 15 feet, and 17 feet.

Question 5

One positive number is 2 larger than another number. The sum of their squares is 100. Find the numbers.
A) 8, 9
B) 6, 8
C) 7, 9
D) 4, 6
E) 9, 10

Accepted Answer

Let the smaller number be x. Then the larger number is x + 2. The equation is x^2 + (x + 2)^2 = 100. Solving this gives x = 6, so the numbers are 6 and 8.

Question 6

Find three negative consecutive integers such that the product of the two smaller integers is 116 more than ten times the largest integer.
A) -10, -9, -8
B) -8, -7, -6
C) -9, -8, -7
D) -6, -5, -4

Accepted Answer

The answer of Find three negative consecutive integers such that...

Question 7

The length of one leg of a right triangle is 7 centimeters more than the length of the other leg. The length of the hypotenuse is 17 centimeters. Find the lengths of the two legs.
A) 6 cm, 13 cm
B) 10 cm, 17 cm
C) 11 cm, 16 cm
D) 8 cm, 15 cm
E) 5 cm, 13 cm

Accepted Answer

Using the Pythagorean theorem, let x be the length of the shorter leg. Then the other leg is x + 7. Solving x^2 + (x + 7)^2 = 17^2 gives x = 8 and x + 7 = 15.

Question 8

Solve the equation. $- x ^ { 2 } - 3 x + 40 = 0$
A) {3, -40}
B) {5, 8}
C) {-5, 8}
D) {-5, -8}
E) {5, -8}

Accepted Answer

The answer of Solve the equation. \(- x ^ {...

Question 9

An apple orchard contains 50 trees. The number of trees per row is five more than the number of rows. Find the number of rows.

Accepted Answer

The answer of An apple orchard contains 50 trees. The...

Question 10

An apple orchard contains 104 trees. The number of trees per row is five more than the number of rows. Find the number of rows.
A) 8
B) 13
C) 15
D) 5
E) 11

Accepted Answer

The answer of An apple orchard contains 104 trees. The...

Question 11

Solve the equation. $x ( x - 10 ) = - 24$
A) {-4, 6}
B) {10, -24}
C) {4, -6}
D) {-4, -6}
E) {4, 6}

Accepted Answer

The answer of Solve the equation. \(x ( x -...

Question 12

Solve the equation. $x ^ { 2 } + 6 x = 7$
A) {1, 7}
B) {-1, -7}
C) {1, -7}
D) {-1, 7}
E) {6, 7}

Accepted Answer

Rearranging the equation, we get $x^2+6x-7=0$. Factoring or using the quadratic formula, we find that the solutions are $x=1$ and $x=-7$. Therefore, the answer is (C) {1, -7}.

Question 13

Find two consecutive positive odd whole numbers whose product is 35.
A) 4, 6
B) 5, 6
C) 5, 7
D) 3, 5
E) 7, 9

Accepted Answer

Let x be the first odd number. Then the second consecutive odd number is x + 2. We know that their product is 35, so we can set up the equation: x(x + 2) = 35. Solving for x, we get x = 5. Therefore, the two consecutive positive odd whole numbers are 5 and 7 (since 5 + 2 = 7), which is choice C.

Question 14

The perimeter of a rectangle is 44 centimeters and the area is 112 square centimeters. Find the length and width of the rectangle.
A) 7 cm, 16 cm
B) 6 cm, 16 cm
C) 8 cm, 14 cm
D) 7 cm, 13 cm
E) 10 cm, 12 cm

Accepted Answer

Let the length of the rectangle be L and let the width be W.

We know that the perimeter (P) of a rectangle is given by: 
P = 2L + 2W

And we know that the area (A) of a rectangle is given by: 
A = L * W

We are given that the perimeter is 44 cm, so we can write: 
2L + 2W = 44

Simplifying this equation, we get: 
L + W = 22

We are also given that the area is 112 cm^2, so we can write: 
L * W = 112

Now we can use substitution to solve for L and W. From the equation L + W = 22, we can write: 
L = 22 - W

Substituting this into the equation L * W = 112, we get: 
(22 - W) * W = 112

Expanding the left side, we get: 
22W - W^2 = 112

Rearranging and simplifying, we get a quadratic equation: 
W^2 - 22W + 112 = 0

We can solve this by factoring or using the quadratic formula. Factoring gives: 
(W - 8) (W - 14) = 0

So W = 8 or W = 14. If W = 8, then L = 22 - W = 14. If W = 14, then L = 22 - W = 8.

Therefore, the possible dimensions of the rectangle are: 
L = 8 cm, W = 14 cm 
or 
L = 14 cm, W = 8 cm

However, we are looking for the length and width of the rectangle, not just the possible dimensions. Therefore, the correct answer is: 
Choice C: 8 cm, 14 cm

Question 15

Find two consecutive positive odd whole numbers whose product is 143.
A) 13, 15
B) 11, 13
C) 11, 12

Accepted Answer

Let x be the first odd number. Then the second odd number is x+2. Since their product is 143, we can write the equation: x(x+2) = 143. Solving for x, we get x = 11. Therefore, the two consecutive odd numbers are 11 and 13, which corresponds to choice B.

Question 16

The perimeter of a rectangle is 32 centimeters and the area is 48 square centimeters. Find the length and width of the rectangle.
A) 6 cm, 10 cm
B) 4 cm, 12 cm
C) 3 cm, 11 cm
D) 3 cm, 13 cm

Accepted Answer

Let the length of the rectangle be $l$ and the width be $w$. We know that $2(l+w) = 32$, or $l + w = 16$, and that $lw = 48$. We can solve for $w$ in the second equation to get $w = \frac{48}{l}$. We can substitute this into the first equation to get $l + \frac{48}{l} = 16$. Multiplying both sides by $l$ gives us $l^2+48=16l$, which we can rearrange to $l^2-16l+48=0$. Factoring gives us $(l-4)(l-12) = 0$, so $l=4$ or $l=12$. We can check that $w$ is positive in both cases, so the dimensions of the rectangle are either 4 cm by 12 cm or 12 cm by 4 cm. Since the question asks for the length and width of the rectangle, not the other way around, the answer is $B$: 4 cm, 12 cm.

Question 17

The length of one leg of a right triangle is 1 centimeter more than the length of the other leg. The length of the hypotenuse is 5 centimeters. Find the lengths of the two legs.
A) 3 cm, 4 cm
B) 1 cm, 2 cm
C) 6 cm, 5 cm
D) 5 cm, 6 cm

Accepted Answer

Let's denote the length of one leg by x and the length of the other leg by x+1.
Using Pythagoras' theorem, we have:
x^2 + (x+1)^2 = 5^2
x^2 + x^2 + 2x + 1 = 25
2x^2 + 2x - 24 = 0
x^2 + x - 12 = 0
(x+4)(x-3) = 0
x = -4 (discard this solution as it is not a positive length) or x = 3

If x = 3, then the other leg is x+1 = 4. These lengths satisfy the condition of being the legs of a right triangle and the hypotenuse being 5 centimeters long. Therefore, the answer is A, 3 cm and 4 cm.

Question 18

Suppose that the width of a certain rectangle is 6 inches less than its length. The area is numerically 21 less than twice the perimeter. Find the length and width of the rectangle.
A) 1 in., 7 in.
B) 5 in., 11 in.
C) 2 in., 8 in.
D) 3 in., 9 in.
E) 8 in., 14 in.

Accepted Answer

The answer of Suppose that the width of a certain...

Question 19

Suppose that one leg of a right triangle is 5 feet shorter than the other leg. The hypotenuse is 5 feet longer than the longer leg. Find the lengths of all three sides of the right triangle.
A) 17 feet, 22 feet, 27 feet
B) 15 feet, 20 feet, 25 feet
C) 13 feet, 21 feet, 26 feet
D) 13 feet, 22 feet, 25 feet

Accepted Answer

Using the Pythagorean theorem, let x be the length of the shorter leg. Then the longer leg is x + 5, and the hypotenuse is x + 10. The equation is x^2 + (x + 5)^2 = (x + 10)^2. Solving this, x = 15, so the sides are 15 feet, 20 feet, and 25 feet.

Question 20

An apple orchard contains 66 trees. The number of trees per row is five more than the number of rows. Find the number of rows.
A) 9
B) 11
C) 3
D) 6

Accepted Answer

The answer of An apple orchard contains 66 trees. The...

Quiz 7: Factoring, Solving Equations and Problem Solving

One positive number is 4 larger than another number. The sum of their squares is 106. Find the numbers.

Find three negative consecutive integers such that the product of the two smaller integers is 142 more than ten times the largest integer.

Suppose that the width of a certain rectangle is 4 inches less than its length. The area is numerically 11 less than twice the perimeter. Find the length and width of the rectangle.

Suppose that one leg of a right triangle is 7 feet shorter than the other leg. The hypotenuse is 2 feet longer than the longer leg. Find the lengths of all three sides of the right triangle.

One positive number is 2 larger than another number. The sum of their squares is 100. Find the numbers.

Find three negative consecutive integers such that the product of the two smaller integers is 116 more than ten times the largest integer.

The length of one leg of a right triangle is 7 centimeters more than the length of the other leg. The length of the hypotenuse is 17 centimeters. Find the lengths of the two legs.

Solve the equation. −x2−3x+40=0- x ^ { 2 } - 3 x + 40 = 0−x2−3x+40=0

An apple orchard contains 50 trees. The number of trees per row is five more than the number of rows. Find the number of rows.

An apple orchard contains 104 trees. The number of trees per row is five more than the number of rows. Find the number of rows.

Solve the equation. x(x−10) =−24x ( x - 10 ) = - 24x(x−10) =−24

Solve the equation. x2+6x=7x ^ { 2 } + 6 x = 7x2+6x=7

Find two consecutive positive odd whole numbers whose product is 35.

The perimeter of a rectangle is 44 centimeters and the area is 112 square centimeters. Find the length and width of the rectangle.

Find two consecutive positive odd whole numbers whose product is 143.

The perimeter of a rectangle is 32 centimeters and the area is 48 square centimeters. Find the length and width of the rectangle.

The length of one leg of a right triangle is 1 centimeter more than the length of the other leg. The length of the hypotenuse is 5 centimeters. Find the lengths of the two legs.

Suppose that the width of a certain rectangle is 6 inches less than its length. The area is numerically 21 less than twice the perimeter. Find the length and width of the rectangle.

Suppose that one leg of a right triangle is 5 feet shorter than the other leg. The hypotenuse is 5 feet longer than the longer leg. Find the lengths of all three sides of the right triangle.

An apple orchard contains 66 trees. The number of trees per row is five more than the number of rows. Find the number of rows.

Solve the equation. $- x ^ { 2 } - 3 x + 40 = 0$

Solve the equation. $x ( x - 10 ) = - 24$

Solve the equation. $x ^ { 2 } + 6 x = 7$