Topic Quiz A.4: Quadratic equations

Question 1 of 10 · Solve quadratic equations

No calculator

Find every value of x that satisfies x^2 = 7x.

Answer choices
Hint

Get 0 on one side first. Is it safe to divide both sides by x?

Question 2 of 10 · Solve quadratic equations

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Two different numbers x satisfy (x + 2)^2 = 81. Enter the smaller one.

x =

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Undo the square first with \pm. Then you will have two small linear equations.

Question 3 of 10 · Write quadratic equations from context

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The product of two consecutive positive odd integers is 143.

What is the smaller integer?

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Odd numbers are 2 apart. If the smaller one is n, what is the next odd number?

Question 4 of 10 · Write quadratic equations from context

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A bakery sells 200 loaves of bread a day at $4 each. The owner estimates that for every $0.50 increase in price, she will sell 10 fewer loaves.

Let x be the number of $0.50 increases. Which expression gives the daily revenue, in dollars?

Answer choices
Hint

Revenue = (price per loaf) × (number of loaves). Write each one in terms of x.

Question 5 of 10 · Solve quadratic equations

What are the solutions of x^2 + 4x - 7 = 0, rounded to the nearest tenth?

Answer choices
Hint

Use x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a} with a = 1. Watch the sign of c inside the square root.

Question 6 of 10 · Solve quadratic equations

No calculator

The table shows some values of y = x^2 + x - 12.

x-4-3-2-10123
y0-6-10-12-12-10-60

Based on the table, what are the solutions of x^2 + x - 12 = 0?

Answer choices
Hint

A solution makes x^2 + x - 12 equal to 0. Look for the columns where y = 0.

Question 7 of 10 · Write quadratic equations from context

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A rectangular pool is 30 feet by 20 feet. A concrete walkway of the same width, x feet, surrounds the pool on all sides. The area of the walkway alone is 336 square feet.

A rectangular pool 30 feet by 20 feet surrounded by a concrete walkway of the same width x on all sides. The area of the walkway alone (not the pool) is 336 square feet.Pool30 ft by 20 ft30 ft20 ftxxWalkway area (shaded border only) = 336 ft²

How wide is the walkway, in feet?

feet

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Walkway area = (area of the big outer rectangle) − (area of the pool). Each outer dimension is the pool dimension plus 2x.

Question 8 of 10 · Write quadratic equations from context

A rectangular parking lot is 20 feet longer than it is wide. Its area is 2,400 square feet.

What is the width of the parking lot?

Answer choices
Hint

Write w(w + 20) = 2400, move everything to one side, and factor. Then read which dimension the question wants.

Question 9 of 10 · Solve quadratic equations

Before using the quadratic formula, the equation x^2 = 6x - 4 must be put in standard form. Which expression, fully simplified, gives its solutions?

Answer choices
Hint

Rewrite in the form ax^2 + bx + c = 0 first. Then use the formula and simplify \sqrt{20}.

Question 10 of 10 · Write quadratic equations from context

A community theater charges $12 per ticket and sells 300 tickets per show. A survey shows that for every $1 increase in the ticket price, the theater will sell 15 fewer tickets.

Let x be the number of $1 increases. Which equation can be used to find the value(s) of x that would bring in $3,825 per show?

Answer choices
Hint

Revenue = (price per ticket) × (number of tickets). Write each one in terms of x.

Formula sheet

Area

Area of a square s = length of a side
A = s^2
Area of a rectangle l = length, w = width
A = lw
Area of a parallelogram b = base, h = height (straight up from the base, not the slanted side)
A = bh
Area of a triangle b = base, h = height
A = \tfrac{1}{2}bh
Area of a trapezoid b_1 and b_2 = the two parallel sides, h = height
A = \tfrac{1}{2}h(b_1 + b_2)
Area of a circle r = radius (half of the diameter); \pi \approx 3.14
A = \pi r^2

Perimeter and circumference

Perimeter of a square s = length of a side
P = 4s
Perimeter of a rectangle l = length, w = width
P = 2l + 2w
Perimeter of a triangle s_1, s_2, s_3 = the three side lengths
P = s_1 + s_2 + s_3
Circumference of a circle the distance around a circle; r = radius, d = diameter, \pi \approx 3.14
C = 2\pi r = \pi d

Surface area (SA) and volume (V)

p = perimeter of the base with area B; \pi \approx 3.14

Rectangular prism l = length, w = width, h = height
SA = 2lw + 2lh + 2wh \qquad V = lwh
Right prism B = area of the base, p = perimeter of the base, h = height
SA = ph + 2B \qquad V = Bh
Cylinder r = radius of the base, h = height
SA = 2\pi rh + 2\pi r^2 \qquad V = \pi r^2 h
Pyramid B = area of the base, p = perimeter of the base, s = slant height, h = height
SA = \tfrac{1}{2}ps + B \qquad V = \tfrac{1}{3}Bh
Cone r = radius of the base, s = slant height, h = height
SA = \pi rs + \pi r^2 \qquad V = \tfrac{1}{3}\pi r^2 h
Sphere r = radius
SA = 4\pi r^2 \qquad V = \tfrac{4}{3}\pi r^3

Data

Mean (average) add all the values of the data set, then divide by how many values there are
\text{mean} = \dfrac{\text{total of the values}}{\text{number of values}}
Median
the middle value when the data are in order; with an even number of values, it is the mean of the two middle values

Algebra

Slope of a line (x_1, y_1) and (x_2, y_2) = two points on the line
m = \dfrac{y_2 - y_1}{x_2 - x_1}
Slope-intercept form of a line m = slope, b = y-intercept (where the line crosses the y-axis)
y = mx + b
Point-slope form of a line m = slope, (x_1, y_1) = a point on the line
y - y_1 = m(x - x_1)
Standard form of a quadratic equation a, b, c are numbers and a \ne 0
y = ax^2 + bx + c
Quadratic formula solves ax^2 + bx + c = 0
x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
Pythagorean theorem in a right triangle: a, b = the two legs, c = the hypotenuse (the longest side, across from the right angle)
a^2 + b^2 = c^2
Simple interest I = interest, P = principal (the amount saved or borrowed), r = rate per year as a decimal, t = time in years
I = Prt
Distance = rate × time d = distance, r = rate (speed), t = time
d = rt
Total cost
\text{total cost} = (\text{number of units}) \times (\text{price per unit})