Miscellaneous

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

Simplify: \(√3 * √12\)

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

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

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

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

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

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

Solve for x: \(4ˣ - 3(2ˣ) + 2 = 0\)

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

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

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

What is the square root of 144?

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

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

Which of the following is an irrational number?

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

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

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

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

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

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

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

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

What is the value of √0?

√(x/y) is equal to

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

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

Solve: \(x² = 49\)

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

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

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

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

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?

Solve: \((log₂x)² - log₂x⁴ + 3 = 0\)

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

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

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

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

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

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

Which of these is the simplest form of √24?

Does √a + √a = √(2a)?

When multiplying √2 • √3, the result is

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

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

Simplify: \(5 / √2\)

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

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

Solve: \(3²ˣ⁺¹ - 10(3ˣ) + 3 = 0\)

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

What is the result of √1?

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

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

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

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

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

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

Solve |x² - 3x + 2| = x - 1.

What is √13 • √13?

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

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

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

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

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

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

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

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

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

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

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

If b² - 4ac is negative, the equation has

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

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

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

Factor: \(2cos²x - 5cosx + 2\)

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

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

Which graph is a U-shaped curve?

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

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

Simplify the radical: \(√72\)

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

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

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

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

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

Which radical is exactly equal to 9?

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

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

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

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

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

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

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

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?

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

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?

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

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

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 is larger: \(2√5 or √20?\)

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

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

Solve: \(3x² - 12x = 0\)

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

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

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

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

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

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

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

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

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

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

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

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

Solve: \(x² = -16\)

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

Between which two whole numbers does √30 lie?

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

Which function has the 'flattest' or widest curve?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Multiply √2 • √3 • √6.

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

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

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

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

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

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

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

Simplify: \(√18 + √50\)

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

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

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

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

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

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

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

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

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

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

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

Solve: \((x - 3)² = 16\)

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

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

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

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

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

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

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

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

Solve: \(|x² - 1| = 0\)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Solve: \(√(x/x-1) + √(x-1/x) = 13/6\)

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

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

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

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

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.

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

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

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