{"id":15794,"date":"2019-07-30T15:36:55","date_gmt":"2019-07-30T15:36:55","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/quadratic-equations-2\/"},"modified":"2021-10-22T01:44:40","modified_gmt":"2021-10-22T01:44:40","slug":"problem-set-quadratic-equations-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/chapter\/problem-set-quadratic-equations-2\/","title":{"raw":"Problem set: Quadratic Equations","rendered":"Problem set: Quadratic Equations"},"content":{"raw":"
\r\n

Problem sET<\/h3>\r\n
\r\n

Verbal<\/h4>\r\n
\r\n
\r\n\r\n1. How do we recognize when an equation is quadratic?\r\n\r\n<\/div>\r\n
\r\n

[reveal-answer q=\"92824\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"92824\"]It is a second-degree equation (the highest variable exponent is 2).[\/hidden-answer]<\/p>\r\n \r\n\r\n2. When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

[reveal-answer q=\"928479\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"928479\"]We want to take advantage of the zero property of multiplication in the fact that if [latex]a\\cdot b=0[\/latex] then it must follow that each factor separately offers a solution to the product being zero:[latex]a=0 \\text{ or } b=0.[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

3. In the quadratic formula, what is the name of the expression under the radical sign and how does it determine the number of and nature of our solutions?<\/p>\r\n[reveal-answer q=\"531992\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"531992\"]It is called a discriminant. Its sign lets us determine the number and the type of solutions the equation has.[\/hidden-answer]\r\n\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

\r\n

Algebraic<\/h4>\r\n

For the following exercises, solve the quadratic equation by factoring.<\/p>\r\n\r\n

\r\n
\r\n

4. [latex]{x}^{2}-9x+18=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"243451\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"243451\"][latex]x=6, x=3[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

5. [latex]6{x}^{2}+17x+5=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"646866\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"646866\"][latex]x=\\frac{-5}{2}, x=\\frac{-1}{3}[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

6. [latex]3{x}^{2}-75=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"413776\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"413776\"][latex]x=5, x=-5[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

7. [latex]4{x}^{2}=9[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

[reveal-answer q=\"989290\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"989290\"][latex]x=\\frac{-3}{2}, x=\\frac{3}{2}[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n\r\n8.\u00a0[latex]<\/span>5{x}^{2}=5x+30[\/latex]\r\n\r\n<\/div>\r\n
\r\n

[reveal-answer q=\"443800\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"443800\"][latex]x=-2, x=3[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

9.\u00a0[latex]<\/span>7{x}^{2}+3x=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"511778\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"511778\"][latex]x=0, x=\\frac{-3}{7}[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\r\n\r\n

\r\n
\r\n

10.\u00a0[latex]{x}^{2}+4x+7=0[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

[reveal-answer q=\"990709\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"990709\"]Not real[\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

11.\u00a0[latex]<\/span>9{x}^{2}-30x+25=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"102848\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"102848\"]One rational[\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

12.\u00a0[latex]<\/span>6{x}^{2}-x-2=0[\/latex]<\/p>\r\n\r\n<\/div>\r\n

\r\n

[reveal-answer q=\"439329\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"439329\"]Two real; rational[\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

For the following exercises, solve the quadratic equation by using the quadratic formula.<\/p>\r\n13.\u00a0[latex]<\/span>{x}^{2}+x=4[\/latex]\r\n

\r\n
\r\n

[reveal-answer q=\"135923\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"135923\"][latex]x=\\frac{-1\u00b1\\sqrt{17}}{2}[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

14.\u00a0\u00a0[latex]<\/span>3{x}^{2}-5x+1=0[\/latex]<\/span><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n

[reveal-answer q=\"355407\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"355407\"][latex]x=\\frac{5\u00b1\\sqrt{13}}{6}[\/latex][\/hidden-answer]<\/p>\r\n \r\n\r\n<\/div>\r\n<\/div>\r\n

\r\n
\r\n\r\n15. [latex]{x}^{2}=-25[\/latex]\r\n\r\n[reveal-answer q=\"45495\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"45495\"][latex]\\left\\{5i,-5i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n16.\u00a0[latex]{x}^{2}+36=0[\/latex]\r\n\r\n[reveal-answer q=\"844737\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"844737\"][latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n17.\u00a0[latex]{x}^{2}+2x+5=0[\/latex]\r\n\r\n[reveal-answer q=\"740613\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"740613\"][latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n18.\u00a0[latex]{x}^{2}+8x+25=0[\/latex]\r\n\r\n[reveal-answer q=\"825675\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"825675\"][latex]\\left\\{2+i,2-i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n19.\u00a0[latex]{x}^{2}+6x+25=0[\/latex]\r\n\r\n[reveal-answer q=\"153376\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"153376\"][latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n20.\u00a0[latex]{x}^{2}-6x+10=0[\/latex]\r\n\r\n[reveal-answer q=\"552316\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"552316\"][latex]\\left\\{5+i,5-i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n21.\u00a0[latex]x\\left(x - 2\\right)=10[\/latex]\r\n\r\n[reveal-answer q=\"177143\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"177143\"][latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n22.\u00a0[latex]5{x}^{2}-8x+5=0[\/latex]\r\n\r\n[reveal-answer q=\"364360\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"364360\"][latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n23. [latex]2{x}^{2}-6x+5=0[\/latex]\r\n\r\n[reveal-answer q=\"674654\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"674654\"][latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex][\/hidden-answer]\r\n\r\n \r\n\r\n24.\u00a0[latex]{x}^{2}-2x+4=0[\/latex]\r\n\r\n[reveal-answer q=\"19775\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"19775\"][latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex][\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"
\n

Problem sET<\/h3>\n
\n

Verbal<\/h4>\n
\n
\n

1. How do we recognize when an equation is quadratic?<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
It is a second-degree equation (the highest variable exponent is 2).<\/div>\n<\/div>\n

 <\/p>\n

2. When we solve a quadratic equation by factoring, why do we move all terms to one side, having zero on the other side?<\/span><\/p>\n<\/div>\n<\/div>\n

\n
\n

\n

Show Answer<\/span><\/p>\n
We want to take advantage of the zero property of multiplication in the fact that if [latex]a\\cdot b=0[\/latex] then it must follow that each factor separately offers a solution to the product being zero:[latex]a=0 \\text{ or } b=0.[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

3. In the quadratic formula, what is the name of the expression under the radical sign and how does it determine the number of and nature of our solutions?<\/p>\n

Show Answer<\/span><\/p>\n
It is called a discriminant. Its sign lets us determine the number and the type of solutions the equation has.<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n<\/div>\n

\n

Algebraic<\/h4>\n

For the following exercises, solve the quadratic equation by factoring.<\/p>\n

\n
\n

4. [latex]{x}^{2}-9x+18=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
[latex]x=6, x=3[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

5. [latex]6{x}^{2}+17x+5=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
[latex]x=\\frac{-5}{2}, x=\\frac{-1}{3}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

6. [latex]3{x}^{2}-75=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
[latex]x=5, x=-5[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

7. [latex]4{x}^{2}=9[\/latex]<\/p>\n<\/div>\n<\/div>\n

\n
\n

\n

Show Answer<\/span><\/p>\n
[latex]x=\\frac{-3}{2}, x=\\frac{3}{2}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

8.\u00a0[latex]<\/span>5{x}^{2}=5x+30[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
[latex]x=-2, x=3[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

9.\u00a0[latex]<\/span>7{x}^{2}+3x=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
[latex]x=0, x=\\frac{-3}{7}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

For the following exercises, determine the discriminant, and then state how many solutions there are and the nature of the solutions. Do not solve.<\/p>\n

\n
\n

10.\u00a0[latex]{x}^{2}+4x+7=0[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

\n
\n

\n

Show Answer<\/span><\/p>\n
Not real<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

11.\u00a0[latex]<\/span>9{x}^{2}-30x+25=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
One rational<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

12.\u00a0[latex]<\/span>6{x}^{2}-x-2=0[\/latex]<\/p>\n<\/div>\n

\n

\n

Show Answer<\/span><\/p>\n
Two real; rational<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

For the following exercises, solve the quadratic equation by using the quadratic formula.<\/p>\n

13.\u00a0[latex]<\/span>{x}^{2}+x=4[\/latex]<\/p>\n

\n
\n

\n

Show Answer<\/span><\/p>\n
[latex]x=\\frac{-1\u00b1\\sqrt{17}}{2}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

14.\u00a0\u00a0[latex]<\/span>3{x}^{2}-5x+1=0[\/latex]<\/span><\/p>\n<\/div>\n<\/div>\n

\n
\n

\n

Show Answer<\/span><\/p>\n
[latex]x=\\frac{5\u00b1\\sqrt{13}}{6}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n<\/div>\n<\/div>\n

\n
\n

15. [latex]{x}^{2}=-25[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{5i,-5i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

16.\u00a0[latex]{x}^{2}+36=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{2i\\sqrt{2},-2i\\sqrt{2}\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

17.\u00a0[latex]{x}^{2}+2x+5=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{3i\\sqrt{3},-3i\\sqrt{3}\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

18.\u00a0[latex]{x}^{2}+8x+25=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{2+i,2-i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

19.\u00a0[latex]{x}^{2}+6x+25=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{2+3i,2 - 3i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

20.\u00a0[latex]{x}^{2}-6x+10=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{5+i,5-i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

21.\u00a0[latex]x\\left(x - 2\\right)=10[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{2+2\\sqrt{6}, 2 - 2\\sqrt{6}\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

22.\u00a0[latex]5{x}^{2}-8x+5=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{-\\frac{1}{2}+\\frac{3}{2}i, -\\frac{1}{2}-\\frac{3}{2}i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

23. [latex]2{x}^{2}-6x+5=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{-\\frac{3}{5}+\\frac{1}{5}i, -\\frac{3}{5}-\\frac{1}{5}i\\right\\}[\/latex]<\/div>\n<\/div>\n

 <\/p>\n

24.\u00a0[latex]{x}^{2}-2x+4=0[\/latex]<\/p>\n

Show Answer<\/span><\/p>\n
[latex]\\left\\{-\\frac{1}{2}+\\frac{1}{2}i\\sqrt{7}, -\\frac{1}{2}-\\frac{1}{2}i\\sqrt{7}\\right\\}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"author":473810,"menu_order":17,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jay-abramson"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[75],"license":[57],"class_list":["post-15794","chapter","type-chapter","status-publish","hentry","contributor-jay-abramson","license-cc-by"],"part":10990,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15794","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/users\/473810"}],"version-history":[{"count":15,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15794\/revisions"}],"predecessor-version":[{"id":15862,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15794\/revisions\/15862"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/parts\/10990"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapters\/15794\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/media?parent=15794"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/pressbooks\/v2\/chapter-type?post=15794"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/contributor?post=15794"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/ccbcmd-math-1\/wp-json\/wp\/v2\/license?post=15794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}