5.6.0: Exercises — Read — Invisible Network

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For the following exercises, multiply the binomials.

Exercise $\PageIndex{1}$

$(w + 5)(w + 7)$

Exercise $\PageIndex{2}$

$(y + 9)(y + 3)$

Exercise $\PageIndex{3}$

$(p + 11)(p - 4)$

Exercise $\PageIndex{4}$

$(q + 4)(q - 8)$

Exercise $\PageIndex{5}$

$(x + 8)(x + 3)$

Exercise $\PageIndex{6}$

$(2x - 1)(x + 6)$

For the following exercises, factor the trinomials.

Exercise $\PageIndex{7}$

${a^2} - 5a - 14$

Exercise $\PageIndex{8}$

${n^2} + 10n + 24$

Exercise $\PageIndex{9}$

${u^2} - 16$

Exercise $\PageIndex{10}$

${x^2} + 4x - 21$

Exercise $\PageIndex{11}$

${y^2} - 8y + 15$

Exercise $\PageIndex{12}$

${x^2} - 25$

For the following exercises, solve the quadratic equations by graphing.

Exercise $\PageIndex{13}$

Graph and list the solutions to the quadratic equation ${x^2} - 5x - 14=0$.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (negative 2, 0) and (7, 0). The parabola is partially visible.$

Exercise $\PageIndex{14}$

Graph and list the solutions to the quadratic equation ${x^2} + 10x + 24=0$.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (negative 6, 0), (negative 5, negative 1), and (negative 4, 0).$

Exercise $\PageIndex{15}$

Graph and list the solutions to the quadratic equation ${x^2} - 25=0$.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (negative 5, 0) and (5, 0). The parabola is partially visible.$

Exercise $\PageIndex{16}$

Find the solutions of $y = {x^2} + 2x - 15$ from its graph.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (negative 5, 0) and (3, 0). The parabola is partially visible.$

Exercise $\PageIndex{17}$

Find the solutions of $y = {x^2} + 6x + 8$ from its graph.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (negative 4, 0), (negative 3, negative 1), and (negative 2, 0).$

Exercise $\PageIndex{18}$

Find the solutions of $y = {x^2} - 9x + 14$ from its graph.

$A parabola is plotted on an x y coordinate plane. The x and y axes range from negative 10 to 10, in increments of 1. The parabola opens up and it passes through the following points, (2, 0), (4.5, negative 6.2), and (7, 0).$

For the following exercises, solve the quadratic equation by factoring.

Exercise $\PageIndex{19}$

${x^2} + 7x + 12 = 0$

Exercise $\PageIndex{20}$

${y^2} - 8y + 15 = 0$

Exercise $\PageIndex{21}$

${n^2} - 5n + 6 = 0$

Exercise $\PageIndex{22}$

${a^2} + 2a = a$

Exercise $\PageIndex{23}$

${b^2} = - b$

Exercise $\PageIndex{24}$

${x^2} - 9x + 14 = 0$

For the following exercises, solve the quadratic equation using the square root method.

Exercise $\PageIndex{25}$

${x^2} = 16$

Exercise $\PageIndex{26}$

$49{m^2} = 144$

Exercise $\PageIndex{27}$

$625 = {x^2}$

For the following exercises, solve the quadratic equation using the quadratic formula.

Exercise $\PageIndex{28}$

$16{p^2} = 24p - 9$

Exercise $\PageIndex{29}$

${m^2} - 2m = 1$

Exercise $\PageIndex{30}$

$4{x^2} - 2x - 3 = 0$

Exercise $\PageIndex{31}$

${y^2} - 6y + 7 = 0$

Exercise $\PageIndex{32}$

${x^2} - 3x - 7 = 0$

Exercise $\PageIndex{33}$

$2{x^2} - 4x - 11 = 0$

Exercise $\PageIndex{34}$

The product of two consecutive odd integers is 99. Find the integers.

Exercise $\PageIndex{35}$

The product of two consecutive even integers is 168. Find the integers.

Exercise $\PageIndex{36}$

A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

For the following exercises, use the Pythagorean Theorem: In any right triangle, where $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypotenuse, as shown in the given figure, ${a^2} + {b^2} = {c^2}$.

$A right triangle. The legs are labeled a and b. The hypotenuse is labeled c$

Exercise $\PageIndex{37}$

Justine wants to put a deck in the corner of her backyard in the shape of a right triangle, as shown in the given figure. The hypotenuse will be 17 feet long. The length of one side will be 7 feet less than the length of the other side. Find the lengths of the sides of the deck.

$A right triangle. The legs are labeled x minus 7 and x. The hypotenuse is labeled 17.$

Exercise $\PageIndex{38}$

A boat’s sail is a right triangle. The length of one side of the sail is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the sail.

Exercise $\PageIndex{39}$

The sun casts a shadow from a flagpole. The height of the flagpole is three times the length of its shadow. The distance between the end of the shadow and the top of the flagpole is 21 feet. Find the length of the shadow and the length of the flagpole. Round to the nearest tenth of a foot.

Exercise $\PageIndex{40}$

Rene is setting up a holiday light display. He wants to make a “tree” in the shape of two right triangles and has two 10-foot strings of lights to use for the sides. He will attach the lights to the top of a pole and to two stakes on the ground. He wants the height of the pole to be the same as the distance from the base of the pole to each stake. How tall should the pole be? Round to the nearest tenth of a foot.

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