Knowledge Check: Chapters 6–10

Simplify: $(12a^2 - 7a + 4) + (3a^2 + 8a - 10)$.

A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $-16t^2 + 1000$ gives the height of the package $t$ seconds after it was dropped. Find the height when $t = 6$ seconds.

Simplify: $\left(-\tfrac{2}{5}\right)^3$.

Simplify: $(4a^3b^5)^2$.

Simplify: $3k(k^2 - 7k + 13)$.

Simplify: $(v - 9)(9v - 5)$.

Simplify: $(n - 6)(n^2 - 5n + 4)$.

Simplify: $(7p - 5)(7p + 5)$.

Square the binomial using the Binomial Squares Pattern: $(8u + 1)^2$.

Simplify: $\tfrac{3^8}{3^{10}}$.

Simplify: $(87x^{15}y^3z^{22})^0$.

Simplify: $\tfrac{12x^2 + 42x - 6}{2x}$.

Simplify: $(64x^3 - 1) \div (4x - 1)$.

Simplify: $5^{-2}$.

Simplify: $q^{-4} \cdot q^{-5}$.

Convert 83,000,000 to scientific notation.

Simplify, and write your answer in decimal form: $(3.4 \times 10^9)(2.2 \times 10^{-5})$.

Find the Greatest Common Factor in the expression: $14y - 42$.

Find the Greatest Common Factor in the expression: $80a^2 + 120a^3$.

Factor completely: $xy - 8y + 7x - 56$.

Factor completely: $ab - 3b - 2a + 6$.

Factor completely: $x^2 + 13x + 42$.

Factor the trinomial: $u^2 + 17u + 72$.

Factor completely: $3a^3 - 6a^2 - 72a$.

Factor completely: $8m^2 + 22m + 5$.

Factor completely: $9s^2 - 12s + 4$.

Factor completely: $100 - a^2$.

Factor: $a^3 - 125$.

Factor completely: $3x^2 - 75y^2$.

Factor completely: $n^4 - 81$.

Solve: $y^2 = y + 132$. Enter the solutions, separated by commas.

Solve: $4n^2 + 19n + 21 = 0$. Enter the solutions, separated by commas.

Simplify: $\tfrac{3a^2b}{6ab^2}$.

Evaluate $\tfrac{4p-1}{p^2+5}$ when $p=-1$.

Multiply: $\tfrac{4x}{x+2}\cdot\tfrac{x^2+5x+6}{12x^2}$.

Divide: $\tfrac{t^2-4t-12}{t^2+8t+12}\div\tfrac{t^2-36}{6t}$.

Add: $\tfrac{p^2+10p}{p+5}+\tfrac{25}{p+5}$.

Subtract: $\tfrac{d^2}{d+4}-\tfrac{3d+28}{d+4}$.

Add: $\tfrac{4}{pq}+\tfrac{5}{p}$.

Add: $\tfrac{2}{c-2}+\tfrac{9}{c+3}$.

Simplify: $\cfrac{\frac{2}{3}+\frac{3}{5}}{\frac{2}{5}}$.

Simplify: $\cfrac{x-\frac{3x}{x+5}}{\frac{1}{x+5}+\frac{1}{x-5}}$.

Solve: $\tfrac{1}{2}+\tfrac{2}{7}=\tfrac{1}{x}$.

Solve: $\tfrac{1}{z-5}+\tfrac{1}{z+5}=\tfrac{1}{z^2-25}$.

Solve: $\tfrac{2}{r-2}=\tfrac{3}{r-1}$.

The recommended erythromycin dosage for dogs, is 5 mg for every pound the dog weighs. If Daisy weighs 25 pounds, how many milligrams of erythromycin should her veterinarian prescribe?

Tony is 5.75 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree.

Kurt can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind's speed is 6 mph, what is Kurt's speed on his bike?

An experienced window washer can wash all the windows in Mike's house in 2 hours, while a new trainee can wash all the windows in 7 hours. How long would it take them working together?

If $y$ varies inversely with $x$ and $x=6$ when $y=20$, find $y$ when $x=2$.

The price that Tyler pays for gas varies directly with the number of gallons he buys. If 24 gallons cost him $59.76, what would 30 gallons cost?

Simplify: $\sqrt{144}$.

Simplify: $\sqrt{64 + 225}$.

Simplify: $\sqrt{169m^4n^2}$.

Simplify: $\sqrt{98}$.

Simplify: $3\sqrt{13} + 5\sqrt{2} + \sqrt{13}$.

Simplify: $5\sqrt{5} + 7\sqrt{5}$.

Simplify: $(3\sqrt{6y})(2\sqrt{50y^3})$.

Simplify: $(1 - 2\sqrt{q})^2$.

Simplify: $\sqrt{\tfrac{14y^3}{7y}}$.

Rationalize the denominator: $\tfrac{3}{2\sqrt{6}}$.

Solve: $3\sqrt{2x - 3} - 20 = 7$.

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula $t = \tfrac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker. Round your answer to the nearest tenth of a second.

Simplify $\sqrt[4]{81x^{12}}$.

Simplify: $\sqrt[4]{512} - 2\sqrt[4]{32}$.

Simplify: $49^{\tfrac{3}{2}}$.

Simplify: $\tfrac{w^{\tfrac{3}{4}}}{w^{\tfrac{7}{4}}}$.

Use the Square Root Property to solve the quadratic equation: $3(w + 5)^2 = 27$. Enter both solutions separated by commas.

Solve using the Square Root Property: $x^2 = 100$. Enter both solutions separated by commas.

Complete the square to make a perfect square trinomial. Then write the result as a binomial squared: $x^2 + 22x$.

Solve by completing the square: $c^2 + 20c = 21$. Enter both solutions separated by commas.

Use the Quadratic Formula to solve the quadratic equation: $2m^2 - 5m + 3 = 0$. Enter both solutions separated by commas.

Solve the quadratic equation using any method: $3n^2 + 8n + 3 = 0$. Enter both solutions separated by commas.

Use the discriminant to determine the number of solutions of the quadratic equation $6p^2 - 13p + 7 = 0$.

Find the two pairs of consecutive even numbers whose product is 360. Enter the four values separated by commas.

A triangular banner has an area of 351 square centimeters. The length of the base is two centimeters longer than four times the height. Find the height and the length of the base. Enter the height followed by the base length, separated by a comma.

For the parabola $y = 3x^2 + 6x + 8$, find the axis of symmetry. Enter the value of $x$.

For the parabola $y = 3x^2 + 6x + 8$, find the vertex.

For the parabola $y = x^2 + 10x + 24$, find the axis of symmetry. Enter the value of $x$.

For the parabola $y = x^2 + 10x + 24$, find the vertex.