Practice Problems

\(\textbf{1)}\) The length of a rectangle is 3 times its width. The area of the rectangle is 48 square yards. Find the dimensions of the rectangle.

Show Answer

\( \text{Width} = 4 \text{ yards}, \text{Length} = 12 \text{ yards} \)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=3w, \, A=48\)
\(\,\,\,\,\, \left(3w\right)\left(w\right)=48\)
\(\,\,\,\,\, 3w^2=48\)
\(\,\,\,\,\, w^2=16\)
\(\,\,\,\,\, w=\pm4\)
\(\,\,\,\,\, w=4\)
\(\,\,\,\,\, l=3(4)=12\)
\(\,\,\,\,\, \text{Width} = 4 \text{ yards}, \text{Length} = 12 \text{ yards} \)

 

\(\textbf{2)}\) The length of a rectangle is 4 times its width. The area of the rectangle is 100 square feet. Find the dimensions of the rectangle.

Show Answer

\( \text{Width} = 5 \text{ ft}, \text{Length} = 20 \text{ ft} \)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=4w, \, A=100\)
\(\,\,\,\,\, \left(4w\right)\left(w\right)=100\)
\(\,\,\,\,\, 4w^2=100\)
\(\,\,\,\,\, w^2=25\)
\(\,\,\,\,\, w=5\)
\(\,\,\,\,\, l=4(w)=4(5)=20\)
\(\,\,\,\,\, \text{Width} = 5 \text{ ft}, \text{Length} = 20 \text{ ft} \)

 

\(\textbf{3)}\) The length of a rectangular plot is 5 ft more than its width. The area of the plot is 66 square ft. Find the dimensions of the plot.

Show Answer

\( \text{Width} = 6 \text{ ft}, \text{Length} = 11 \text{ ft} \)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=w+5, \, A=66\)
\(\,\,\,\,\, (w+5)w=66\)
\(\,\,\,\,\, w^2+5w-66=0\)
\(\,\,\,\,\, (w+11)(w-6)=0\)
\(\,\,\,\,\, w=-11 \text{ or } w=6\)
\(\,\,\,\,\, l=6+5=11 \text{ or } l=-11+5=-6\)
\(\,\,\,\,\, \text{Width} = 6 \text{ ft}, \text{Length} = 11 \text{ ft} \)

 

\(\textbf{4)}\) The length of a rectangle is 15 feet less than its width. The area of the rectangle is 126 square feet. Find the dimensions of the rectangle.

Show Answer

\( \text{Width} = 6 \text{ ft}, \text{Length} = 21 \text{ ft} \)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=w-15, \, A=126\)
\(\,\,\,\,\, (w-15)w=126\)
\(\,\,\,\,\, w^2-15w-126=0\)
\(\,\,\,\,\, (w-21)(w+6)=0\)
\(\,\,\,\,\, w=21 \text{ or } w=-6\)
\(\,\,\,\,\, l=21-15=6 \text{ or } l=-6-15=-21\)
\(\,\,\,\,\, \text{Width} = 21 \text{ ft}, \text{Length} = 6 \text{ ft} \)

 

\(\textbf{5)}\) The length of a rectangle is 3 inches more than double the width. The area of the rectangle is 230 square inches. Find the dimensions of the rectangle.

Show Answer

\(\text{Length} = 23 \text{ in}, \text{Width} = 10 \text{ in} \)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=2w+3, \, A=230\)
\(\,\,\,\,\, (2w+3)w=230\)
\(\,\,\,\,\, 2w^2+3w-230=0\)
\(\,\,\,\,\, (2w+23)(w-10)=0\)
\(\,\,\,\,\, w=-23/2 \text{ or } w=10\)
\(\,\,\,\,\, l=2(10)+3=23 \text{ or } l=2(-23/2)+3=-23+3=-20\)
\(\,\,\,\,\, \text{Length} = 23 \text{ in}, \text{Width} = 10 \text{ in} \)

 

\(\textbf{6)}\) The length of a rectangle is 5 meters more than triple the width. The area is 138 square meters. Find the dimensions of the rectangle.

Show Answer

\(\text{Width} = 6 \text{ m}, \text{Length} = 23 \text{ m}\)

Show Work

\(\,\,\,\,\, \text{Length } (l) \, \times \, \text{Width } (w) = \text{Area } (A)\)
\(\,\,\,\,\, l=3w+5, \, A=138\)
\(\,\,\,\,\, (3w+5)w=138\)
\(\,\,\,\,\, 3w^2+5w-138=0\)
\(\,\,\,\,\, (3w+23)(w-6)=0\)
\(\,\,\,\,\, w=-23/3 \text{ or } w=6\)
\(\,\,\,\,\, l=3(6)+5=18+5=23 \text{ or } l=3(-23/3)+5=-23+5=-18\)
\(\,\,\,\,\, \text{Width} = 6 \text{ m}, \text{Length} = 23 \text{ m}\)

 

\(\textbf{7)}\) The width of a rectangle is 6 meters less than its length. The area is 72 square meters. Find the dimensions of the rectangle.

Show Answer

The answer is \( W = 6, L = 12 \)

Show Work

\(\,\,\,\,\, \text{Length } (L) \, \times \, \text{Width } (W) = \text{Area } (A)\)
\(\,\,\,\,\, L = W + 6, \, A = 72\)
\(\,\,\,\,\, (W + 6)W = 72\)
\(\,\,\,\,\, W^2 + 6W – 72 = 0\)
\(\,\,\,\,\, (W + 12)(W – 6) = 0\)
\(\,\,\,\,\, W = -12 \text{ or } W = 6\)
\(\,\,\,\,\, L = 6 + 6 = 12 \text{ or } L = -12 + 6 = -6\)
\(\,\,\,\,\, \text{Width} = 6 \text{ meters}, \text{Length} = 12 \text{ meters} \)

 

\(\textbf{8)}\) The length of a rectangle is twice the width. The area is 32 square inches. Find the dimensions of the rectangle.

Show Answer

The answer is \( W=4, L=8 \)

Show Work

\(\,\,\,\,\, \text{Length } (L) \, \times \, \text{Width } (W) = \text{Area } (A)\)
\(\,\,\,\,\, L = 2W, \, A = 32\)
\(\,\,\,\,\, (2W)W = 32\)
\(\,\,\,\,\, 2W^2 = 32\)
\(\,\,\,\,\, W^2 = 16\)
\(\,\,\,\,\, W = 4\)
\(\,\,\,\,\, L = 2(4) = 8\)
\(\,\,\,\,\, \text{Width} = 4 \text{ inches}, \text{Length} = 8 \text{ inches} \)

 

\(\textbf{9)}\) The length of a rectangle is 1 foot less than twice the width. The area is 120 square feet. Find the dimensions of the rectangle.

Show Answer

The answer is \( W = 8, L = 15 \)

Show Work

\(\,\,\,\,\, \text{Length } (L) \, \times \, \text{Width } (W) = \text{Area } (A)\)
\(\,\,\,\,\, L = 2W – 1, \, A = 120\)
\(\,\,\,\,\, (2W – 1)W = 120\)
\(\,\,\,\,\, 2W^2 – W – 120 = 0\)
\(\,\,\,\,\, (2W + 15)(W – 8) = 0\)
\(\,\,\,\,\, W = -15/2 \text{ or } W = 8\)
\(\,\,\,\,\, L = 2(8) – 1 = 16 – 1 = 15 \text{ or } L = 2(-15/2) – 1 = -15 – 1 = -16\)
\(\,\,\,\,\, \text{Width} = 8 \text{ feet}, \text{Length} = 15 \text{ feet} \)

 

\(\textbf{10)}\) The product of two positive consecutive integers is 56. Find the integers.

Show Answer

The integers are \( 7, 8 \)

Show Work

Let’s call the first integer “n” and the second integer “n + 1”.
\(\,\,\,\,\, n(n + 1) = 56\)
\(\,\,\,\,\, n^2 + n – 56 = 0\)
\(\,\,\,\,\, (n + 8)(n – 7) = 0\)
\(\,\,\,\,\, n = -8 \text{ or } n = 7\)
Since we’re looking for positive integers, the integers are \(7, 8\)

 

\(\textbf{11)}\) The product of two positive consecutive odd integers is 99. Find the integers.

Show Answer

The integers are \( 9, 11 \)

Show Work

Let’s call the first odd integer “n” and the second odd integer “n + 2”.
\(\,\,\,\,\, n(n + 2) = 99\)
\(\,\,\,\,\, n^2 + 2n – 99 = 0\)
\(\,\,\,\,\, (n + 11)(n – 9) = 0\)
\(\,\,\,\,\, n = -11 \text{ or } n = 9\)
Since we’re looking for positive odd integers, the integers are \(9, 11\)

 

\(\textbf{12)}\) The product of two positive consecutive odd integers is 1 less than 3 times their sum. Find the integers.

Show Answer

The integers are \( 5, 7 \)

Show Work

Let’s call the first odd integer “n” and the second odd integer “n + 2”.
\(\,\,\,\,\, n(n + 2) = 3(n + n + 2) – 1\)
\(\,\,\,\,\, n^2 + 2n = 6n + 6 – 1\)
\(\,\,\,\,\, n^2 – 4n – 5 = 0\)
\(\,\,\,\,\, (n – 5)(n + 1) = 0\)
\(\,\,\,\,\, n = 5 \text{ or } n = -1\)
Since we’re looking for positive odd integers, the integers are \(5, 7\)

 

\(\textbf{13)}\) The product of two positive consecutive integers is thirteen less than five times their sum. Find the integers.

Show Answer

The answer is \( 8, 9 \)

Show Work

Let’s call the first integer “n” and the second integer “n + 1”.
\(\,\,\,\,\, n(n + 1) = 5(n + n + 1) – 13\)
\(\,\,\,\,\, n^2 + n = 10n + 5 – 13\)
\(\,\,\,\,\, n^2 – 9n – 8 = 0\)
\(\,\,\,\,\, (n – 8)(n + 1) = 0\)
\(\,\,\,\,\, n = 8 \text{ or } n = -1\)
Since we’re looking for positive integers, the integers are \(8, 9\)

 

\(\textbf{14)}\) The product of two positive consecutive odd integers is 77 more than twice the larger. Find the integers.

Show Answer

The integers are \( 9, 11 \)

Show Work

Let’s call the first odd integer “n” and the second odd integer “n + 2”.
\(\,\,\,\,\, n(n + 2) = 2(n + 2) + 77\)
\(\,\,\,\,\, n^2 + 2n = 2n + 4 + 77\)
\(\,\,\,\,\, n^2 – 75 = 0\)
\(\,\,\,\,\, (n – 5)(n + 5) = 0\)
\(\,\,\,\,\, n = 5 \text{ or } n = -5\)
Since we’re looking for positive odd integers, the integers are \(5, 7\)

 

Challenge Problem

\(\textbf{15)}\) The width is twice the height. If the perimeter is 120, what is the area?

Show Area

The area is \( 800\) units\(^2 \)

 

 

See Related Pages\(\)

\(\bullet\text{ Adding and Subtracting Polynomials}\)
\(\,\,\,\,\,\,\,\,(4d+7)−(2d−5)…\)

\(\bullet\text{ Multiplying Polynomials}\)
\(\,\,\,\,\,\,\,\,(x+2)(x^2+3x−5)…\)

\(\bullet\text{ Dividing Polynomials}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Dividing Polynomials (Synthetic Division)}\)
\(\,\,\,\,\,\,\,\,(x^3-8)÷(x-2)…\)

\(\bullet\text{ Synthetic Substitution}\)
\(\,\,\,\,\,\,\,\,f(x)=4x^4−3x^2+8x−2…\)

\(\bullet\text{ End Behavior}\)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow \infty, \quad f(x)\rightarrow \infty \)
\(\,\,\,\,\,\,\,\, \text{As } x\rightarrow -\infty, \quad f(x)\rightarrow \infty… \)

\(\bullet\text{ Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+10x−24=0…\)

\(\bullet\text{ Quadratic Formula and the Discriminant}\)
\(\,\,\,\,\,\,\,\,x=-b \pm \displaystyle\frac{\sqrt{b^2-4ac}}{2a}…\)

\(\bullet\text{ Complex Numbers}\)
\(\,\,\,\,\,\,\,\,i=\sqrt{-1}…\)

\(\bullet\text{ Multiplicity of Roots}\)
\(\,\,\,\,\,\,\,\,\)\(…\)

\(\bullet\text{ Rational Zero Theorem}\)
\(\,\,\,\,\,\,\,\, \pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12…\)

\(\bullet\text{ Descartes Rule of Signs}\)
\(\,\)

\(\bullet\text{ Roots and Zeroes}\)
\(\,\,\,\,\,\,\,\,\text{Solve for }x. 3x^2+4x=0…\)

\(\bullet\text{ Linear Factored Form}\)
\(\,\,\,\,\,\,\,\,f(x)=(x+4)(x+1)(x−3)…\)

\(\bullet\text{ Polynomial Inequalities}\)
\(\,\,\,\,\,\,\,\,x^3-4x^2-4x+16 \gt 0…\)

 

In Summary

Doing quadratic word problems is a great way to reinforce the algebra behind quadratic equations. It also adds a fun element by demonstrating real-world applications.

<!–

\(\textbf{10)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

10. The length of a rectangle is 2 less than three times the width. Find the dimensions of the rectangle if the
area is 65 square meters.

ANS:W = 5, L = 13

\(\textbf{11)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

11. The length of a rectangle is 7 meters less than twice the width. Find the dimensions if the area is 60 square
meters.

ANS:W = 7.5, L = 8

\(\textbf{14)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

14. Find two consecutive even integers such that the square of the smaller is 10 more than the larger.

ANS:4 & 6

\(\textbf{17)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

17. The product of two consecutive even integers is 6 more than three times their sum. Find the integers.

ANS:-2, 0 & 6, 8

\(\textbf{19)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

19. The product of two consecutive integers is 5 more than three times the larger. Find the integers.

ANS:4, 5 & -2, -1

\(\textbf{20)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

20. Find three consecutive integers such that four times the sum of all three is 2 times the product of the larger
two.

ANS:4, 5, 6 & -1, 0, 1

\(\textbf{21)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

21. Find three consecutive integers such that three times the sum of all three equals the product of the larger
two.

ANS:7, 8, 9 & -1, 0, 1

\(\textbf{22)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

22. The medium side of a right triangle is 7 more than the shortest side. The longest side is 7 less than 3 times
the shortest side. Find the length of the shortest side of the triangle.

ANS: 8

\(\textbf{23)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

23. One leg of a right triangle is one inch shorter than the other leg. If the hypotenuse is 5 inches, find the
length of the shorter leg.

ANS:3 inches

\(\textbf{24)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

24. The longer leg of a right triangle is two inches more than twice the length of the shorter leg. The
hypotenuse is two inches less than three times the length of the shorter leg. Find the length of the
hypotenuse.

ANS:13 inches

\(\textbf{25)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

25. The longer leg of a right triangle is ten less than three times the shorter leg. The hypotenuse is 4 more than
the shorter leg. Find the length of the shorter leg.

ANS:6

\(\textbf{26)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

26. The hypotenuse of a right triangle is 3 less than twice the shorter leg. The length of the other leg is 3 more
than the shorter leg. Find the length of the shorter leg.

ANS:9

\(\textbf{27)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

27. The hypotenuse of a right triangle is 1 centimeter longer than the longer leg. The shorter leg is 7
centimeters shorter than the longer leg. Find the length of the longer leg of the triangle.

ANS:12

\(\textbf{28)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

28. The longer leg of a right triangle is 1 meter longer than the shorter leg. The hypotenuse is 1 meter shorter
than twice the shorter leg. Find the length of the shorter leg of the triangle.

ANS: 3

\(\textbf{29)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

29. A ladder is resting against a wall. The top of the ladder touches the wall at a height of 15 feet. Find the
distance from the wall to the bottom of the ladder if the length of the ladder is one foot more than twice its
distance from the wall.

ANS: 8 feet

\(\textbf{30)}\) Words \(123\)

Show Answer

The answer is \( x=3 \)

 

30. Two cars leave an intersection. One car travels north; the other travels east. When the car traveling north
had gone 24 miles, the distance between the cars was four miles more than three times the distance
traveled by the car heading east. Find the distance between the cars at that time.

ANS: 25 miles