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Mathematics LibreTexts

7.6E: Exercises

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    30447
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    Practice Makes Perfect

    Use the Zero Product Property

    In the following exercises, solve.

    Exercise 1

    \((x−3)(x+7)=0\)

    Answer

    \(x=3\), \(x=−7\) so the solution set is: \(\{3, -7\}\)

    Exercise 2

    \((y−11)(y+1)=0\)

    Exercise 3

    \((3a−10)(2a−7)=0\)

    Answer

    \(a=\frac{10}{3}\), \(a=\frac{7}{2}\) so the solution set is: \(\Big\{\tfrac{10}{3}, \tfrac{7}{2}\Big\}\)

    Exercise 4

    \((5b+1)(6b+1)=0\)

    Exercise 5

    \(6m(12m−5)=0\)

    Answer

    \(m=0\), \(m=\frac{5}{12}\) so the solution set is: \(\Big\{0, \tfrac{5}{12}\Big\}\)

    Exercise 6

    \(2x(6x−3)=0\)

    Exercise 7

    \((y−3)^2=0\)

    Answer

    \(y=3\) so the solution set is: \(\{3\}\)

    Exercise 8

    \((b+10)^2=0\)

    Exercise 9

    \((2x−1)^2=0\)

    Answer

    \(x=\frac{1}{2}\) so the solution set is: \(\Big\{\tfrac{1}{2}\Big\}\)

    Exercise 10

    \((3y+5)^2=0\)

    Solve Quadratic Equations by Factoring

    In the following exercises, solve.

    Exercise 11

    \(x^2+7x+12=0\)

    Answer

    \(x=−3\), \(x=−4\) so the solution set is: \(\{-3, -4\}\)

    Exercise 12

    \(y^2−8y+15=0\)

    Exercise 13

    \(5a^2−26a=24\)

    Answer

    \(a=−\tfrac{4}{5}\), \(a=6\) so the solution set is: \(\Big\{−\tfrac{4}{5}, 6\Big\}\)

    Exercise 14

    \(4b^2+7b=−3\)

    Exercise 15

    \(4m^2=17m−15\)

    Answer

    \(m=\frac{5}{4}\), \(m=3\) so the solution set is: \(\Big\{\tfrac{5}{4}, 3\Big\}\)

    Exercise 16

    \(n^2=5−6n\)

    Exercise 17

    \(7a^2+14a=7a\)

    Answer

    \(a=−1\), \(a=0\) so the solution set is: \(\{-1, 0\}\)

    Exercise 18

    \(12b^2−15b=−9b\)

    Exercise 19

    \(49m^2=144\)

    Answer

    \(m=\frac{12}{7}\), \(m=−\frac{12}{7}\) so the solution set is: \(\Big\{−\tfrac{12}{7}, \tfrac{12}{7}\Big\}\)

    Exercise 20

    \(625=x^2\)

    Exercise 21

    \((y−3)(y+2)=4y\)

    Answer

    \(y=−1\), \(y=6\) so the solution set is: \(\{-1, 6\}\)

    Exercise 22

    \((p−5)(p+3)=−7\)

    Exercise 23

    \((2x+1)(x−3)=−4x\)

    Answer

    \(x=\frac{3}{2}\), \(x=−1\) so the solution set is: \(\Big\{-1,\tfrac{3}{2}\Big\}\)

    Exercise 24

    \((x+6)(x−3)=−8\)

    Exercise 25

    \(16p^3=24p^2−9p\)

    Answer

    \(p=0\), \(p=\frac{3}{4}\) so the solution set is: \(\Big\{0,\tfrac{3}{4}\Big\}\)

    Exercise 26

    \(m^3−2m^2=−m\)

    Exercise 27

    \(20x^2−60x=−45\)

    Answer

    \(x=\frac{3}{2}\) so the solution set is: \(\Big\{\tfrac{3}{2}\Big\}\)

    Exercise 28

    \(3y^2−18y=−27\)

    ​​​​​Solve Applications Modeled by Quadratic Equations

    In the following exercises, solve.

    Exercise 29

    The product of two consecutive integers is 56. Find the integers.

    Answer

    7 and 8; −8 and −7

    Exercise 30

    The product of two consecutive integers is 42. Find the integers.

    Exercise 31

    The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

    Answer

    4 feet and 7 feet

    Exercise 32

    A rectangular retaining wall has area 15 square feet. The height of the wall is two feet less than its length. Find the height and the length of the wall.

    Exercise 33

    A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

    Answer

    6 feet and 8 feet

    Exercise 34

    A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

    Mixed Practice

    In the following exercises, solve.

    Exercise 35

    (x+8)(x−3)=0

    Answer

    \(x=−8, \; x=3\) so the solution set is: \(\{-8, 3\}\)

    Exercise 36

    (3y−5)(y+7)=0

    Exercise 37

    \(p^2+12p+11=0\)

    Answer

    \(p=−1, \;p=−11\) so the solution set is: \(\{-11, -1\}\)

    Exercise 38

    \(q^2−12q−13=0\)

    Exercise 39

    \(m^2=6m+16\)

    Answer

    \(m=−2, \; m=8\) so the solution set is: \(\{-2, 8\}\)

    Exercise 40

    \(4n^2+19n=5\)

    Exercise 41

    \(a^3−a^2−42a=0\)

    Answer

    \(a=0, \;a=−6, \;a=7\) so the solution set is: \(\{-6, 0, 7\}\)

    Exercise 42

    \(4b^2−60b+224=0\)

    Exercise 43

    The product of two consecutive integers is 110. Find the integers.

    Answer

    10 and 11; −11 and −10

    Exercise 44

    The length of one leg of a right triangle is three more than the other leg. If the hypotenuse is 15, find the lengths of the two legs.

    Everyday Math

    Exercise 45

    Area of a patio If each side of a square patio is increased by 4 feet, the area of the patio would be 196 square feet. Solve the equation (s+4)2=196(s+4)2=196 for s to find the length of a side of the patio.

    Answer

    10 feet

    Exercise 46

    Watermelon drop A watermelon is dropped from the tenth story of a building. Solve the equation −16t2+144=0−16t2+144=0 for tt to find the number of seconds it takes the watermelon to reach the ground.

    Writing Exercises

    Exercise 47

    Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

    Answer

    Answers may vary for the explanation.  You should expect no more than 2 solutions for a quadratic equation.  Often it has two solutions, but sometimes, it can have one repeated solution or even no solution.

    Exercise 48

    Give an example of a quadratic equation that has a GCF and none of the solutions to the equation is zero.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has the following statements all to be preceded by “I can…”. The first row is “solve quadratic equations by using the zero product property”. The second row is “solve quadratic equations by factoring”. The third row is “solve applications modeled by quadratic equations”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

    b. Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

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