{"questions":[{"uuid":"e9ee8749-ce1a-45bb-bcc1-d4b750d7f831","question":"Solve the quadratic equation

x^2 - 4x - 5 = 0

. What are the roots?","answer":"x = 5, x = -1","alternatives":["x = -5, x = -1","x = -5, x = 1","x = 1, x = 5"],"answerExplanation":"To find the roots of

x^2 - 4x - 5 = 0

, we use the quadratic formula

x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}

. Plugging in

a = 1

b = -4

, and

c = -5

, we get

x = \\frac{4 \\pm \\sqrt{16 + 20}}{2}

which simplifies to

x = 5

and

x = -1

.","hint":"Consider using the quadratic formula:

x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}

.","imageEnabled":false,"duration":30,"latexEnabled":false,"audioEnabled":false,"htmlEnabled":true,"questionId":"e9ee8749-ce1a-45bb-bcc1-d4b750d7f831"},{"uuid":"fa3bdef6-e136-4d97-997f-2ce54116428f","question":"Simplify the expression

\\sqrt{50}

.","answer":"5\\sqrt{2}","alternatives":["\\sqrt{500}","2\\sqrt{25}","10\\sqrt{5}"],"answerExplanation":"

\\sqrt{50}

can be simplified by recognizing that

50 = 25 \\times 2

. Therefore,

\\sqrt{50} = \\sqrt{25 \\times 2} = \\sqrt{25} \\times \\sqrt{2} = 5\\sqrt{2}

.","hint":"Factor the radicand and look for perfect squares.","imageEnabled":false,"duration":30,"latexEnabled":false,"audioEnabled":false,"htmlEnabled":true,"questionId":"fa3bdef6-e136-4d97-997f-2ce54116428f"},{"uuid":"81319f8e-bc91-4c66-8277-5a3bb42ba7a0","question":"Solve the quadratic equation

2x^2 + 3x - 2 = 0

. What are the roots?","answer":"x = \\frac{1}{2}, x = -2","alternatives":["x = \\frac{-1}{2}, x = 2","x = 1, x = -2","x = -1, x = 2"],"answerExplanation":"To find the roots of

2x^2 + 3x - 2 = 0

, use the quadratic formula:

x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}

. Here,

a = 2

b = 3

, and

c = -2

. Substituting these values, we get

x = \\frac{-3 \\pm \\sqrt{3^2 - 4 \\cdot 2 \\cdot (-2)}}{2 \\cdot 2}

which simplifies to

x = \\frac{-3 \\pm \\sqrt{9 + 16}}{4} = \\frac{-3 \\pm 5}{4}

. Therefore, the roots are

x = \\frac{1}{2}

and

x = -2

.","hint":"Use the quadratic formula

x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}

.","imageEnabled":false,"duration":30,"latexEnabled":false,"audioEnabled":false,"htmlEnabled":true,"questionId":"81319f8e-bc91-4c66-8277-5a3bb42ba7a0"},{"uuid":"607fe356-18e6-45ad-afb7-015ee2bf5b67","question":"Simplify the expression

\\sqrt{72} - \\sqrt{18}

.","answer":"3\\sqrt{2}","alternatives":["\\sqrt{54}","6\\sqrt{2}","2\\sqrt{3}"],"answerExplanation":"First, simplify each radical separately.

\\sqrt{72} = \\sqrt{36 \\times 2} = 6\\sqrt{2}

and

\\sqrt{18} = \\sqrt{9 \\times 2} = 3\\sqrt{2}

. Then, subtract the simplified radicals:

6\\sqrt{2} - 3\\sqrt{2} = 3\\sqrt{2}

.","hint":"Break down each radical to its simplest form first.","imageEnabled":false,"duration":30,"latexEnabled":false,"audioEnabled":false,"htmlEnabled":true,"questionId":"607fe356-18e6-45ad-afb7-015ee2bf5b67"}]}

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