Miscellaneous

A rectangle has a perimeter of 20. If width is 'x', then length is (10 - x). The area equation is

The lowest or highest point on a parabola is specifically called the

In the equation 3x² - 5 = 2x, what is the value of 'b' for the quadratic formula?

Consider a parabola with a vertex at (0,5) opening upward and a line y = 2. How many times do they intersect?

What is √13 • √13?

Simplify √48. (Hint: \(16 is a factor).\)

Simplify √(1/4). This is the same as √1 / √4, which equals

A 5x7 garden is surrounded by a walkway of width 'x'. The total area is (5+2x)(7+2x). Why is it '2x' instead of 'x'?

Which function has the 'flattest' or widest curve?

To simplify √75, we factor 75 into 25 and 3. The square root of 25 is 5, so the answer is

Simplify: \(√18 + √50\)

A car travels North 'x' miles and East 'x+7' miles. The straight-line distance squared is 169. Equation

Solve: \((x - 5)(x + 2) = 0\)

True or False: \((√a + √b)² = a + b.\)

A group rents a bus for $400. If 5 more people join, the cost per person drops by $4. This setup is

Why do we 'rationalize the denominator' for a fraction like 1/√3?

To solve the system y = x² - 4 and y = 5 graphically, you should look for the x-values where x² - 4 equals

To find the initial height of a projectile from its equation, you should look at the value when t is

When solving y = x² + 2 and y = 3x, the first step is setting them equal. This represents finding the points where

Simplify the radical: \(√72\)

If a system of a line and parabola has exactly one solution, the line is said to be ___ to the parabola.

A system has solutions at (2, 4) and (-2, 4). This could be the intersection of y = x² and

Simplify √(x³). (Hint: \(This can be written as √(x² • x)).\)

When simplifying √20, we can write it as √(4 • 5). Since √4 is 2, the final form is

If you have 4√x and add 1√x, how many √x do you have in total?

To complete the square for x² + 10x, what number must be added?

Solve for x: \(x^(2/3) - 5x^(1/3) + 6 = 0\)

Simplify √(25x²). Assuming x is positive, the result is

To complete the square for 2x² + 8x, what is the first step?

Can the width of a physical garden ever be a negative solution from a quadratic equation?

If a ball takes 3 seconds to reach its peak, how many total seconds until it hits the ground (if thrown from ground)?

A line with a positive slope passes through the vertex of a parabola opening upward. What is the minimum number of intersections?

Simplify 10√7 - 4√7. Just like 10x - 4x, the answer is

Revenue is (Price × Quantity). If price is 'p' and quantity is (50 - p), the revenue equation is

If you square a square root, such as (√x)², the result is simply

Step 1: \(Simplify √8 into 2√2. Step 2\)

Which point lies on the graph of y = 2x² - 1?

Find the sum of roots for |x - 2|² + |x - 2| - 6 = 0.

Solve: \(x² - 5x + 2√(x² - 5x + 3) = 12. (Hint\)

Which is larger: \(2√5 or √20?\)

Simplify √12 + √3. First turn √12 into 2√3. Now add them

To solve x⁴ + 2x³ - 6x² + 2x + 1 = 0, we divide by x² and substitute y = x + 1/x. What is the resulting quadratic in y?

If the x-intercepts of a parabola are at x=2 and x=8, what is the x-coordinate of the vertex?

To find the y-intercept of any quadratic graph, you should set x equal to

Simplify √98. (Hint: \(98 = 49 • 2).\)

A table shows points (-1, 1), (0, 0), and (1, 1). This most likely represents which shape?

Solve: \((log₂x)² - log₂x⁴ + 3 = 0\)

Simplify 10√6 / 2√3. Divide the outside (10/2) and the inside (√6/√3).

In the expression √45, the number 45 is specifically called the

After rationalizing 3/√2, the new numerator will be

In the motion formula h = -16t² + vt + s, what does the 's' represent?

How many solutions can a system of two different parabolas have?

If a square has side 's', the square of its diagonal 'd' is given by s² + s² = d². This simplifies to

How many real solutions exist for |x² - 4| = 3?

Solve: \(3²ˣ⁺¹ - 10(3ˣ) + 3 = 0\)

Which factor of 32 should be used to simplify √32?

For the graph of y = ax² + bx + c, the vertical line of symmetry is always x =

A parabola opens upward. The y-coordinate of its vertex represents the

What is the vertex of the function y = (x - 5)² + 10?

What is the result of √1?

Solve √(x + 7) = x + 1. Which of the following is the only valid solution?

A square's area is doubled when its side is increased by 2. This leads to (s+2)² = 2s². Solve for the original side s.

If you solve a quadratic word problem for 'time' and get t = 5 and t = -2, which answer do you keep?

Which equation represents a horizontal shift of y = x² followed by a vertical stretch?

The graph of y = (x - 2)² is the same as y = (2 - x)².

Which change to y = x² creates y = x² + k (where k > 0)?

What substitution is typically used for the equation 2x⁴ + x³ - 6x² + x + 2 = 0?

Simplify √200. (Hint: \(100 is the largest perfect square factor).\)

Two pipes fill a tank. If one takes 'x' hours and the other takes 'x+1' hours, the combined rate involves which type of equation?

A 20-inch wire is cut into two pieces, x and (20-x). If each is bent into a square, the area of the first square is

If the side of a square is tripled (3s), the area becomes how many times larger?

The product of two consecutive integers is 72. Equation: \(x(x + 1) = 72. In standard form, this is\)

Two numbers have a sum of 12. If one is 'x', the other is (12 - x). Their product is

To rationalize 1 / (2 + √3), we multiply by the conjugate, which is

When no number is written in the 'crook' of the radical (the index), it is understood to be

Does the system y = x² + 10 and y = x - 5 have any real intersection points?

The Product Property of Square Roots states that √(a • b) is equal to

Which radical is exactly equal to 9?

Identify the equation for y = x² shifted 2 units left and 3 units down.

Which of these is the simplest form of √24?

The hypotenuse of a right triangle is 10. One leg is x, the other is x+2. Which setup is correct?

Which of the following is an irrational number?

A 4x6 photo has a frame of uniform width 'x'. The area of the photo AND frame is

Find the values of x² for the equation 9x⁴ - 37x² + 4 = 0.

When multiplying √2 • √3, the result is

The graph of y = -x² is a reflection of y = x² across the

What is the value of √0?

Length is 5 less than twice the width. If width is 'w', length is

Can a linear equation and a quadratic equation intersect 0 times?

When a projectile returns to the same height it started from, the displacement is

If the vertex of a parabola is (3, -4) and it opens upward, how many x-intercepts must it have?

When a word problem asks when an object 'hits the ground,' what value should you plug in for height (h)?

A ball is thrown. h(t) = -16t² + 32t. To find when the ball returns to the height it was thrown from (0), you solve for the

The sum of the squares of two consecutive integers is 25. Which equation is correct?

The area of a rectangle is 50. The length is twice the width (w). What is the equation?

If time 't' is in seconds and height 'h' is in feet, the velocity 'v' is measured in

Solve: \(√(x/x-1) + √(x-1/x) = 13/6\)

Factor: \(2cos²x - 5cosx + 2\)

Find the real roots of x⁴ - 5x² + 4 = 0.

If an object is 'dropped' (not thrown), the initial velocity 'v' in h = -16t² + vt + s is

The sum of a number and its reciprocal is 2.5. This leads to the equation x + 1/x = 2.5. Multiplying by x gives

Solve: \((x - 3)² = 16\)

Which of these radicals does NOT need to be simplified further?

If the vertex of a parabola is (3, -4) and it opens downward, how many x-intercepts must it have?

What is the range of the function y = -(x-1)² + 5?

Factor x - 5√x + 6 = 0 into linear-root factors.

To simplify a square root, we look for the largest factor of the number that is a

If Profit = -x² + 10x - 16, 'breaking even' means the profit is 0. This is found by finding the

If 'x' is the first of two consecutive odd integers, how do you represent the second one?

A quadratic equation has a discriminant of 0. How many x-intercepts does its graph have?

Find the discriminant of x² + 5x + 4 = 0.

Simplify √x⁶ assuming x is positive. (Hint: \(What times itself equals x⁶?)\)

If |a| < 1 in the equation y = ax², the parabola is

Solve: \(3x² - 12x = 0\)

Simplify: \(√3 * √12\)

In the expression 7√2, the number 7 is called the

Solve: \(|x² - 1| = 0\)

Multiply √2 • √3 • √6.

Why does squaring √(2x - 1) = -5 yield an extraneous solution?

Which graph will have a peak (maximum) rather than a valley (minimum)?

A 10x20 pool has a deck of width 'x'. The area of the deck ONLY is Total Area minus Pool Area. This is

Multiply (2√3) • (4√5). (Hint: \(Multiply the outside numbers, then the inside numbers).\)

A rectangular yard uses a house as one side. If you have 40ft of fence and the width is 'x', the length is (40 - 2x). The Area is

A path is h = -x² + 4x + 5. The horizontal distance 'x' where the max height occurs is -b/2a. Here, x is

The square of a number 'n' is equal to 10 more than 3 times the number. Which equation fits?

Solve for x: \(4ˣ - 3(2ˣ) + 2 = 0\)

In the standard equation y = x² + 6x + 5, which constant represents the point where the graph crosses the vertical axis?

Solve: \(x² = 49\)

Does √a + √a = √(2a)?

The area of a triangle is 20. If the base is x and height is (x+2), the equation is 20 = 0.5(x)(x+2). What is the first step to clear the fraction?

As x increases from -10 to 0 for the graph y = x², the y-values

What is (3√2)²? (Hint: \(Square the 3 and square the √2, then multiply).\)

A student solves a system of a line and a parabola and finds 3 intersection points. Is this possible?

Solve the system: \( y = x² y = x + 2 \)

In the formula h = -16t² + s, why is the '16' negative?

Simplify √50 / √2 by first dividing the numbers inside the radical. This becomes

If b² - 4ac is negative, the equation has

Which graph is a U-shaped curve?

To move the graph of y = x² three units to the right, how should the equation be modified?

A ball is at ground level at t=0 and t=4. At what time does it reach its max height?

What is the square root of 144?

The length of a rug is 3 feet more than its width (w). Which expression represents the area?

To simplify √(9/16), you can take the square root of the numerator and denominator separately. This results in

A store sells 100 items at $10. For every $1 increase (x), they sell 5 fewer items. The new quantity is

A ball is dropped from 64 feet. Its height is h = -16t² + 64. When does it hit the ground?

If a parabola's vertex is at (3, 4) and it opens upward, how many real x-intercepts does it have?

If (1, 5) is a point on a parabola with axis of symmetry x=3, which other point must be on the graph?

Compared to y = x², the graph of y = 4x² will appear

In the function y = (x + 4)², the '+4' moves the graph toward which side of the x-axis?

A system of y = x² and a horizontal line has no solution. Where must the line be located?

On the graph of y = x², for which x-values is the graph 'falling' or decreasing?

Solve using Quadratic Formula: \(x² - 4x - 1 = 0\)

Where do y = x² and y = -x² intersect?

If the area of a circle is 25π, then the equation πr² = 25π allows us to cancel π and find r is

How does the graph of y = x² + 5 compare to y = x² - 5?

For the graph of y = x² + 2, which of these is a possible y-value?

In y = a(x-h)²+k, if 'a' is doubled, what happens to the vertex?

A ladder leans against a wall. The height it reaches and the distance from the wall are 'a' and 'b'. To find the ladder length 'c', we use

If h = -16t² + 100, what is the height after 1 second?

√(x/y) is equal to

Which of these can be added directly to 3√5 because they are 'like radicals'?

Between which two whole numbers does √30 lie?

Simplify: \(5 / √2\)

The vertical line that cuts a parabola into two matching halves is called the

To rationalize 2/√5, you should multiply both the top and the bottom by

If a line is tangent to a parabola, how many solutions does the system have?

In the function y = x² - 7, the vertex is located on which axis?

On a graph, the solutions to a quadratic equation are the points where the parabola crosses the

In the vertex form y = a(x - h)² + k, the value of 'k' tells you how far the parabola has shifted

While a square root asks 'what times itself', a cube root (∛) asks what number used as a factor ___ times equals the radicand?

If x + 1/x = 3, find the value of x² + 1/x².

If the leading coefficient 'a' in y = ax² + bx + c is negative, the 'arms' of the parabola will point

A rocket's height follows h = -16t² + 64t. At t = 0, what is the height of the rocket?

In standard form y = ax² + bx + c, which letter determines if the graph opens up or down?

To find the price that generates the MOST revenue, you need to find the ___ of the revenue parabola.

True or False: \(√9 + √16 is equal to √25.\)

In h = -5t² + 20t (using meters), the '-5' is half of the acceleration of gravity on Earth. Is this true?

Simplify √2(3 + √2). (Hint: \(Multiply √2 by 3, then √2 by √2).\)

A sail's height is 4ft more than its base. Area is 30. Using A = 1/2 bh, the equation is

Solve |x² - 3x + 2| = x - 1.

To find the maximum height an object reaches, you should find the y-value of the

If a ball hits the ground at t = 2.456 seconds, 'to the nearest tenth' would be

In h = -16t² + 40t + 5, the object was thrown upward with an initial velocity of

Solve: \(x² = -16\)

Solving x² = 2x + 3 graphically involves finding the intersection of y = x² and

Squaring a number and taking the square root are ___ operations.

In the set of real numbers, why is √-16 considered 'not a real number'?

Find two numbers whose difference is 4 and product is 45. If the smaller is 'x', the equation is

Step 1: \(Find the vertex. Step 2\)

If a parabola's vertex sits exactly on the x-axis, how many real roots does it have?

An object falls from 100ft. The equation is 0 = -16t² + 100. To solve for 't', the first step is to move 100 to the other side and then

Find the real roots of x⁴ + 3x² - 4 = 0.