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Nonlinear equations and systems quiz

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What is the value of $$y$$?"}],["$","ul",null,{"children":[["$","li","0",{"children":"77"}]]}]]}]]}]]}],["$","$L18",null,{"quiz":{"id":2318,"prismaId":"cmpz9rk8m0e84jgllczfdw11o","slug":"psat-8-9-nonlinear-equations-systems-of-equations-quiz-3","title":{"rendered":"Quiz 3"},"content":{"rendered":"

Practice on nonlinear equations and systems of equations.

"},"excerpt":{"rendered":"

Nonlinear equations and systems quiz

"},"explanation":null,"metaTitle":null,"metaDescription":null,"featured_media":0,"featured_media_url":"$undefined","acf":{"duree_estimee":"$undefined","niveau_difficulte":"Advanced","categorie":"Advanced Math: Nonlinear Equations & Systems of Equations","nombre_questions":10,"score_minimum":80,"ordre_questions":"Fixe","questions":[{"id":"cmrk8eyyf0x0rjg8wzg2v466z","texte_question":"

Given the system:

$$x^2+y+5=5$$

$$10x+20-y=0$$

What is a possible value of $$x$$?

","type_question":"QCM","explication":"

Start with the first equation. $$x^2+y+5=5$$ Subtract $$5$$ from both sides. $$x^2+y=0$$ Solve for $$y$$. $$y=-x^2$$ Now use the second equation. $$10x+20-y=0$$ Solve for $$y$$. $$y=10x+20$$ Set the two expressions for $$y$$ equal. $$-x^2=10x+20$$ Move all terms to one side. $$x^2+10x+20=0$$ Use the quadratic formula. $$x=\\frac{-10\\pm\\sqrt{10^2-4(1)(20)}}{2(1)}$$ Simplify. $$x=\\frac{-10\\pm\\sqrt{100-80}}{2}$$ $$x=\\frac{-10\\pm\\sqrt{20}}{2}$$ $$x=\\frac{-10\\pm2\\sqrt{5}}{2}$$ $$x=-5\\pm\\sqrt{5}$$ Final answer: $$-5-\\sqrt{5}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$-5$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$-2$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$5$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$-5-\\sqrt{5}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyi0x0xjg8wsh3azjl6","texte_question":"

The volume $$V_B$$ of a liquid mixture is given by:

$$V_B=\\frac{V_P}{1-H}$$

where $$V_P$$ is a component volume and $$H$$ is a fraction. Which expression correctly solves for $$H$$?

","type_question":"QCM","explication":"

Start with the equation. $$V_B=\\frac{V_P}{1-H}$$ Multiply both sides by $$1-H$$. $$V_B(1-H)=V_P$$ Divide both sides by $$V_B$$. $$1-H=\\frac{V_P}{V_B}$$ Subtract $$1$$ from both sides. $$-H=\\frac{V_P}{V_B}-1$$ Multiply both sides by $$-1$$. $$H=1-\\frac{V_P}{V_B}$$ Final answer: $$H=1-\\frac{V_P}{V_B}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$H=1-\\frac{V_P}{V_B}$$","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"$$$H=\\frac{V_B}{V_P}$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$H=1+\\frac{V_B}{V_P}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$H=V_B-V_P$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyl0x13jg8wi5xzaf03","texte_question":"

Given $$36x^2-(12a+6b)x+ab=0$$, the sum of the solutions is $$k(2a+b)$$. What is the value of $$k$$?

","type_question":"QCM","explication":"

For a quadratic equation: $$Ax^2+Bx+C=0$$ The sum of the solutions is: $$-\\frac{B}{A}$$ For the equation: $$36x^2-(12a+6b)x+ab=0$$ The value of $$A$$ is: $$36$$ The value of $$B$$ is: $$-(12a+6b)$$ So the sum of the solutions is: $$-\\frac{-(12a+6b)}{36}$$ Simplify. $$\\frac{12a+6b}{36}$$ Factor the numerator. $$\\frac{6(2a+b)}{36}$$ Simplify. $$\\frac{2a+b}{6}$$ The sum is given as: $$k(2a+b)$$ So: $$k=\\frac{1}{6}$$ Final answer: $$\\frac{1}{6}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$\\frac{1}{6}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyo0x16jg8wqyzqpu55","texte_question":"

Given the system:

$$x^2+y+8=8$$

$$6x+12-y=0$$ The solution to the system is $$(x,y)$$. What is the value of $$x$$?

","type_question":"QCM","explication":"

Start with the first equation. $$x^2+y+8=8$$ Subtract $$8$$ from both sides. $$x^2+y=0$$ Solve for $$y$$. $$y=-x^2$$ Now use the second equation. $$6x+12-y=0$$ Solve for $$y$$. $$y=6x+12$$ Set the two expressions for $$y$$ equal. $$-x^2=6x+12$$ Move all terms to one side. $$x^2+6x+12=0$$ Use the quadratic formula. $$x=\\frac{-6\\pm\\sqrt{6^2-4(1)(12)}}{2(1)}$$ Simplify. $$x=\\frac{-6\\pm\\sqrt{36-48}}{2}$$ $$x=\\frac{-6\\pm\\sqrt{-12}}{2}$$ Since the discriminant is negative, there is no real value of $$x$$. Final answer: no real solution.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$-6$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$-2$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$2$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

No real solution

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyr0x1cjg8wwqut024g","texte_question":"

The solutions to $$x^2+4x+3=0$$ are $$r$$ and $$s$$. The solutions to $$x^2+6x+5=0$$ are $$t$$ and $$u$$. The solutions to $$x^2+10x+c=0$$ are $$r+t$$ and $$s+u$$. What is the value of $$c$$?

","type_question":"TexteLibre","explication":"

Factor the first equation. $$x^2+4x+3=0$$ $$(x+1)(x+3)=0$$ So the solutions are: $$-1$$ and $$-3$$ Factor the second equation. $$x^2+6x+5=0$$ $$(x+1)(x+5)=0$$ So the solutions are: $$-1$$ and $$-5$$ Use matching order. $$r=-3$$ $$s=-1$$ $$t=-5$$ $$u=-1$$ Now find the new solutions. $$r+t=-3+(-5)$$ $$r+t=-8$$ $$s+u=-1+(-1)$$ $$s+u=-2$$ The equation $$x^2+10x+c=0$$ has solutions $$-8$$ and $$-2$$. The constant term is the product of the solutions. $$c=(-8)(-2)$$ $$c=16$$ Final answer: $$16$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyu0x1fjg8wqmk8g2q9","texte_question":"

A line with equation $$2y=c$$ intersects the parabola $$y=-x^2+6x$$ at exactly one point. What is the value of $$c$$?

","type_question":"TexteLibre","explication":"

The line is: $$2y=c$$ Solve for $$y$$. $$y=\\frac{c}{2}$$ This is a horizontal line. For the horizontal line to intersect the parabola at exactly one point, it must pass through the vertex of the parabola. The parabola is: $$y=-x^2+6x$$ Use the vertex formula. $$x=\\frac{-b}{2a}$$ Here: $$a=-1$$ $$b=6$$ Substitute. $$x=\\frac{-6}{2(-1)}$$ $$x=3$$ Find the y-value at $$x=3$$. $$y=-(3)^2+6(3)$$ $$y=-9+18$$ $$y=9$$ So the horizontal line is: $$y=9$$ Since: $$y=\\frac{c}{2}$$ Set: $$\\frac{c}{2}=9$$ Multiply by $$2$$. $$c=18$$ Final answer: $$18$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyx0x1ijg8wpwshbmw3","texte_question":"

Given $$(x+1)(x+4)=(x-1)(x-2)+6$$, which of the following is a solution?

","type_question":"QCM","explication":"

Expand the left side. $$(x+1)(x+4)=x^2+5x+4$$ Expand the right side. $$(x-1)(x-2)+6=x^2-3x+2+6$$ Simplify. $$(x-1)(x-2)+6=x^2-3x+8$$ Set the expressions equal. $$x^2+5x+4=x^2-3x+8$$ Subtract $$x^2$$ from both sides. $$5x+4=-3x+8$$ Add $$3x$$ to both sides. $$8x+4=8$$ Subtract $$4$$. $$8x=4$$ Divide by $$8$$. $$x=\\frac{1}{2}$$ Final answer: $$\\frac{1}{2}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$1$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$2$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$3$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{1}{2}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyyz0x1ojg8wr193ywf5","texte_question":"

Given the system:

$$y=x+2$$

$$y=x^2+2$$

Which could be a value of $$x$$?

","type_question":"QCM","explication":"

Set the two expressions for $$y$$ equal. $$x+2=x^2+2$$ Subtract $$2$$ from both sides. $$x=x^2$$ Move all terms to one side. $$x^2-x=0$$ Factor. $$x(x-1)=0$$ Set each factor equal to $$0$$. $$x=0$$ or: $$x=1$$ So a possible value of $$x$$ is $$0$$ or $$1$$. Final answer: $$1$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$-2$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$1$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$2$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$-1$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyz20x1ujg8wyb4qo928","texte_question":"

Given the system:

$$y=x^2-6x+9$$

$$y=9-x$$

If $$(x,y)$$ satisfies the system, what is one possible value of $$x$$?

","type_question":"QCM","explication":"

Set the two expressions for $$y$$ equal. $$x^2-6x+9=9-x$$ Subtract $$9$$ from both sides. $$x^2-6x=-x$$ Add $$x$$ to both sides. $$x^2-5x=0$$ Factor. $$x(x-5)=0$$ Set each factor equal to $$0$$. $$x=0$$ or: $$x=5$$ Final answer: $$5$$

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

$$1$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$5$$

","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$7$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$9$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8eyz50x20jg8w20ddhg19","texte_question":"

Given the system:

$$y=x^2+10x+21$$

$$x+5=9$$

The solution to the system is $$(x,y)$$. What is the value of $$y$$?

","type_question":"TexteLibre","explication":"

Start with the second equation. $$x+5=9$$ Subtract $$5$$ from both sides. $$x=4$$ Substitute $$x=4$$ into the first equation. $$y=x^2+10x+21$$ $$y=4^2+10(4)+21$$ Simplify. $$y=16+40+21$$ $$y=77$$ Final answer: $$77$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"

","correcte":true,"explication":"","imageUrl":"$undefined"}]}]},"categories":["advanced-math-nonlinear-equations-and-systems-of-equations"],"date":"2026-06-04T09:04:47.158Z","modified":"2026-07-14T05:49:52.114Z"}}],false,["$","div",null,{"className":"mt-10 mb-12","aria-label":"Advertisement","children":["$","div",null,{"className":"my-8 ","children":["$","$L19",null,{"adSlot":"1234567890","adFormat":"auto","fullWidthResponsive":true,"style":{"minHeight":"250px"}}]}]}]]}]}]]

Advanced Math: Nonlinear Equations & Systems of Equations — Quiz 3

Questions in this quiz (10)