7:[["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"Quiz\",\"name\":\"Quiz 4\",\"description\":\"Nonlinear equations and systems quiz\",\"url\":\"https://theschoolofmathematics.com/quiz/psat-8-9-nonlinear-equations-systems-of-equations-quiz-4\",\"educationalLevel\":\"Advanced\",\"numberOfQuestions\":10,\"isAccessibleForFree\":true,\"about\":{\"@type\":\"Thing\",\"name\":\"Advanced Math: Nonlinear Equations & Systems of Equations\"}}"}}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"BreadcrumbList\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https://theschoolofmathematics.com\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Quizzes\",\"item\":\"https://theschoolofmathematics.com/quiz\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Quiz 4\",\"item\":\"https://theschoolofmathematics.com/quiz/psat-8-9-nonlinear-equations-systems-of-equations-quiz-4\"}]}"}}],["$","div",null,{"className":"bg-gradient-to-b from-white via-gray-50 to-white","children":["$","div",null,{"className":"container mx-auto px-4 py-8 md:py-12","children":[["$","div",null,{"className":"mb-12","children":["$undefined",["$","div",null,{"className":"mb-8","children":["$","h1",null,{"className":"text-4xl md:text-5xl lg:text-6xl font-bold text-gray-900 mb-4","children":"Quiz 4"}]}],["$","div",null,{"className":"bg-white rounded-2xl shadow-lg p-6 md:p-8 border border-gray-200","children":[["$","div",null,{"className":"text-lg md:text-xl text-gray-700 mb-8 leading-relaxed prose prose-lg max-w-none","dangerouslySetInnerHTML":{"__html":"

Nonlinear equations and systems quiz

"}}],["$","div",null,{"className":"grid grid-cols-2 md:grid-cols-3 gap-4","children":["$undefined",["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M8.228 9c.549-1.165 2.03-2 3.772-2 2.21 0 4 1.343 4 3 0 1.4-1.278 2.575-3.006 2.907-.542.104-.994.54-.994 1.093m0 3h.01M21 12a9 9 0 11-18 0 9 9 0 0118 0z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Questions"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":10}]]}]]}],["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M9 12l2 2 4-4m6 2a9 9 0 11-18 0 9 9 0 0118 0z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Level"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":"Advanced"}]]}]]}],["$","div",null,{"className":"flex items-center gap-3 p-4 bg-gray-50 rounded-xl border border-gray-200","children":[["$","div",null,{"className":"w-10 h-10 bg-gray-900 rounded-lg flex items-center justify-center flex-shrink-0","children":["$","svg",null,{"className":"w-5 h-5 text-white","fill":"none","stroke":"currentColor","viewBox":"0 0 24 24","children":["$","path",null,{"strokeLinecap":"round","strokeLinejoin":"round","strokeWidth":2,"d":"M7 7h.01M7 3h5c.512 0 1.024.195 1.414.586l7 7a2 2 0 010 2.828l-7 7a2 2 0 01-2.828 0l-7-7A1.994 1.994 0 013 12V7a4 4 0 014-4z"}]}]}],["$","div",null,{"children":[["$","div",null,{"className":"text-xs text-gray-500 font-medium","children":"Category"}],["$","div",null,{"className":"text-sm font-bold text-gray-900","children":"Advanced Math: Nonlinear Equations & Systems of Equations"}]]}]]}]]}]]}]]}],["$","$L17",null,{"quiz":{"id":2507,"prismaId":"cmpz9rkal0e9jjgllvq6qyupa","slug":"psat-8-9-nonlinear-equations-systems-of-equations-quiz-4","title":{"rendered":"Quiz 4"},"content":{"rendered":"

Practice on nonlinear equations and systems of equations.

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

Nonlinear equations and systems quiz

"},"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":"cmrk8ez050x24jg8w2gn3zz7i","texte_question":"

Given the system:

$$y=-3$$

$$y=x^2+6x+k$$

In the system above, $$k$$ is a positive integer. The system has no real solutions. What is the least possible value of $$k$$?

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

Set the two expressions for $$y$$ equal. $$-3=x^2+6x+k$$ Move all terms to one side. $$0=x^2+6x+k+3$$ For the system to have no real solutions, the quadratic must have no real roots. That means the discriminant must be less than $$0$$. $$b^2-4ac$$<$$0$$ Here: $$a=1$$ $$b=6$$ $$c=k+3$$ Substitute. $$6^2-4(1)(k+3)$$<$$0$$ Simplify. $$36-4k-12$$<$$0$$ $$24-4k$$<$$0$$ Subtract $$24$$. $$-4k$$<$$-24$$ Divide by $$-4$$ and reverse the inequality. $$k$$>$$6$$ The least positive integer greater than $$6$$ is: $$7$$ Final answer: $$7$$.

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

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez080x27jg8wonqegs9u","texte_question":"

Given $$45x^2+(9b+a)x+ab=0$$, where $$a$$ and $$b$$ are positive constants. The product of the solutions is $$kab$$, where $$k$$ is a constant. What is the value of $$k$$?

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

For a quadratic equation: $$Ax^2+Bx+C=0$$ The product of the solutions is: $$\\frac{C}{A}$$ For the equation: $$45x^2+(9b+a)x+ab=0$$ The value of $$A$$ is: $$45$$ The value of $$C$$ is: $$ab$$ So the product of the solutions is: $$\\frac{ab}{45}$$ The product is given as: $$kab$$ So: $$kab=\\frac{ab}{45}$$ Divide both sides by $$ab$$. $$k=\\frac{1}{45}$$ Final answer: $$\\frac{1}{45}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$\\frac{1}{45}$$","correcte":true,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\frac{1}{9}$$

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

$$1$$

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

$$45$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0b0x2djg8wrgpivnqs","texte_question":"

Given the system:

$$x-y=2$$

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

Which ordered pair is a solution?

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

Start with the first equation. $$x-y=2$$ Solve for $$y$$. $$y=x-2$$ Substitute $$y=x-2$$ into the second equation. $$x+y=x^2-6$$ $$x+(x-2)=x^2-6$$ Simplify. $$2x-2=x^2-6$$ Move all terms to one side. $$0=x^2-2x-4$$ Use the quadratic formula. $$x=\\frac{2\\pm\\sqrt{(-2)^2-4(1)(-4)}}{2}$$ Simplify. $$x=\\frac{2\\pm\\sqrt{4+16}}{2}$$ $$x=\\frac{2\\pm\\sqrt{20}}{2}$$ $$x=1\\pm\\sqrt{5}$$ Since $$y=x-2$$: If $$x=1+\\sqrt{5}$$, then: $$y=-1+\\sqrt{5}$$ One solution is: $$(1+\\sqrt{5},-1+\\sqrt{5})$$ Final answer: $$(1+\\sqrt{5},-1+\\sqrt{5})$$.

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

$$(\\sqrt{2},-\\sqrt{2})$$

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

$$(2+\\sqrt{6},\\sqrt{6})$$

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

$$(\\sqrt{6},-2+\\sqrt{6})$$

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

$$(1+\\sqrt{5},-1+\\sqrt{5})$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0e0x2jjg8w6m4owwxa","texte_question":"

Given $$3x^2-12x-5=0$$, one solution can be written as $$\\frac{12-\\sqrt{k}}{6}$$. What is the value of $$k$$?

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

Use the quadratic formula. For: $$3x^2-12x-5=0$$ The values are: $$a=3$$ $$b=-12$$ $$c=-5$$ Substitute into the discriminant. $$b^2-4ac=(-12)^2-4(3)(-5)$$ Simplify. $$144+60=204$$ So one solution is: $$x=\\frac{12-\\sqrt{204}}{6}$$ Therefore: $$k=204$$ Final answer: $$204$$.

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

204

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0h0x2mjg8wqw6j8lyv","texte_question":"

The speed of an object is modeled by $$v=200+0.5T$$, where $$T$$ is temperature. Which expression correctly solves for $$T$$?

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

Start with the equation. $$v=200+0.5T$$ Subtract $$200$$ from both sides. $$v-200=0.5T$$ Divide both sides by $$0.5$$. $$T=\\frac{v-200}{0.5}$$ Final answer: $$T=\\frac{v-200}{0.5}$$.

","points":1,"temps_limite":"$undefined","reponses":[{"texte":"$$$T=\\frac{v+0.5}{200}$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$T=\\frac{v-0.5}{200}$$

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

$$T=\\frac{v+200}{0.5}$$

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

$$T=\\frac{v-200}{0.5}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0k0x2sjg8w2c89k5or","texte_question":"

Given the system:

$$x+y=20$$

$$xy=96$$

If $$(x,y)$$ is a solution, what is one possible value of $$x$$?

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

The system gives the sum and product of $$x$$ and $$y$$. $$x+y=20$$ $$xy=96$$ So $$x$$ and $$y$$ are the two numbers whose sum is $$20$$ and product is $$96$$. Set up the quadratic equation. $$t^2-20t+96=0$$ Factor. $$(t-12)(t-8)=0$$ So: $$t=12$$ or: $$t=8$$ Therefore, one possible value of $$x$$ is $$8$$ or $$12$$. Final answer: $$12$$

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

$$12$$

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

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0n0x2wjg8wrrke9qo2","texte_question":"

Given the system:

$$y+k=x+10$$

$$y-k=x^2-4x$$

The system has exactly one solution. What is the value of $$k$$?

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

Solve each equation for $$y$$. From the first equation: $$y+k=x+10$$ $$y=x+10-k$$ From the second equation: $$y-k=x^2-4x$$ $$y=x^2-4x+k$$ Set the two expressions equal. $$x+10-k=x^2-4x+k$$ Move all terms to one side. $$0=x^2-5x+2k-10$$ For the system to have exactly one solution, the discriminant must equal $$0$$. $$b^2-4ac=0$$ Here: $$a=1$$ $$b=-5$$ $$c=2k-10$$ Substitute. $$(-5)^2-4(1)(2k-10)=0$$ Simplify. $$25-8k+40=0$$ $$65-8k=0$$ Solve for $$k$$. $$8k=65$$ $$k=\\frac{65}{8}$$ Final answer: $$\\frac{65}{8}$$.

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

$$\\frac{65}{8}$$

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

$$\\frac{5}{8}$$

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

$$\\frac{1}{8}$$

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

$$\\frac{1}{4}$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0r0x32jg8wjlixyj4r","texte_question":"

Given $$x^2-4x-5=0$$, one solution can be written as $$2+\\sqrt{k}$$. What is the value of $$k$$?

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

Factor the equation. $$x^2-4x-5=0$$ $$(x-5)(x+1)=0$$ So the solutions are: $$x=5$$ and: $$x=-1$$ One solution is written as: $$2+\\sqrt{k}$$ Set it equal to $$5$$. $$2+\\sqrt{k}=5$$ Subtract $$2$$. $$\\sqrt{k}=3$$ Square both sides. $$k=9$$ Final answer: $$9$$.

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

$$1$$

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

$$4$$

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

$$9$$

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

$$16$$

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0u0x38jg8wx4ii55re","texte_question":"

Given $$\\frac{10x}{5y}=2\\sqrt{w+9}$$, which expression represents $$w$$ in terms of $$x$$ and $$y$$?

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

Start with the equation. $$\\frac{10x}{5y}=2\\sqrt{w+9}$$ Simplify the left side. $$\\frac{10x}{5y}=\\frac{2x}{y}$$ So: $$\\frac{2x}{y}=2\\sqrt{w+9}$$ Divide both sides by $$2$$. $$\\frac{x}{y}=\\sqrt{w+9}$$ Square both sides. $$\\left(\\frac{x}{y}\\right)^2=w+9$$ Subtract $$9$$ from both sides. $$w=\\left(\\frac{x}{y}\\right)^2-9$$ Final answer: $$\\left(\\frac{x}{y}\\right)^2-9$$.

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

$$\\left(-\\frac{y}{x}\\right)^2+3$$.

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

$$\\left(\\frac{x}{y}\\right)^2-9$$

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

$$\\left(\\frac{y}{x}\\right)^2+9$$.

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

$$\\left(-\\frac{x}{y}\\right)^2-3$$.

","correcte":false,"explication":"","imageUrl":"$undefined"}]},{"id":"cmrk8ez0x0x3ejg8w0z8ocrgo","texte_question":"

Given $$x^2=-49$$, how many distinct real solutions does the equation have?

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

Start with the equation. $$x^2=-49$$ For any real number $$x$$, the value of $$x^2$$ is always greater than or equal to $$0$$. So $$x^2$$ cannot equal a negative number. Since $$-49$$ is negative, there are no real solutions. Final answer: $$0$$.

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

$$\\text{Exactly one}$$

","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$\\text{Exactly two}$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"$$$\\text{Infinitely many}$$","correcte":false,"explication":"","imageUrl":"$undefined"},{"texte":"

$$\\text{Zero}$$

","correcte":true,"explication":"","imageUrl":"$undefined"}]}]},"categories":["advanced-math-nonlinear-equations-and-systems-of-equations"],"date":"2026-06-04T09:04:47.229Z","modified":"2026-07-14T05:49:52.176Z"}}],["$","div",null,{"className":"container mx-auto px-4 pb-12","children":["$","$L18",null,{"targetType":"quiz","targetSlug":"psat-8-9-nonlinear-equations-systems-of-equations-quiz-4"}]}]]}]}]]

Quiz 4

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