x=(-6-sqrt(6^2-96))/(2*3)

{ "query": { "display": "$$x=\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot\\:3}$$", "symbolab_question": "EQUATION#x=\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot 3}" }, "solution": { "level": "PERFORMED", "subject": "Algebra", "topic": "Equations", "subTopic": "Linear", "default": "x=-1-i\\sqrt{\\frac{5}{3}}", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "$$x=\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot\\:3}{\\quad:\\quad}x=-1-i\\sqrt{\\frac{5}{3}}$$", "input": "x=\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot\\:3}", "steps": [ { "type": "interim", "title": "Simplify $$\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot\\:3}:{\\quad}-1-i\\sqrt{\\frac{5}{3}}$$", "input": "\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot\\:3}", "steps": [ { "type": "interim", "title": "$$-6-\\sqrt{6^{2}-96}=-6-\\sqrt{60}i$$", "input": "-6-\\sqrt{6^{2}-96}", "steps": [ { "type": "interim", "title": "$$\\sqrt{6^{2}-96}=\\sqrt{60}i$$", "input": "\\sqrt{6^{2}-96}", "steps": [ { "type": "step", "primary": "$$6^{2}=36$$", "result": "=\\sqrt{36-96}" }, { "type": "step", "primary": "Subtract the numbers: $$36-96=-60$$", "result": "=\\sqrt{-60}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt{-a}=\\sqrt{-1}\\sqrt{a}$$", "secondary": [ "$$\\sqrt{-60}=\\sqrt{-1}\\sqrt{60}$$" ], "result": "=\\sqrt{-1}\\sqrt{60}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply imaginary number rule: $$\\sqrt{-1}=i$$", "result": "=\\sqrt{60}i", "meta": { "practiceLink": "/practice/complex-numbers-practice", "practiceTopic": "Expand FOIL" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7Z/sExgEwnyJLQ5GHTf3gUSDMRVfzkDSnK0l/twHy2CX9ovYKijQYhJDCbxu/nAOJEnTiWsGZAdb3jHBNEG2/mcxkARoM44WPcMutmKWsbuwC+j7pT6Q/XELq3GWgoTLReOJHjp7Fp7EZLzPxUh4DeA==" } }, { "type": "step", "result": "=-6-\\sqrt{60}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7H0yykuvpCWzhqlVdSEZ+PG4Eqzauo3vVY88SHnKDbFBwkKGJWEPFPk38sdJMsyPIYkYq4RMQqPfu+MV31ELT7D/L0MoYg+CUn6oyL3EO7YqFY1xOZDGtv5oLoz5UbFWPJ7QUE9NP4rVU73IQz1ROM5FwDo8zrjModjKQeRvKL2I=" } }, { "type": "step", "result": "=\\frac{-6-\\sqrt{60}i}{2\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:3=6$$", "result": "=\\frac{-6-\\sqrt{60}i}{6}" }, { "type": "interim", "title": "$$\\sqrt{60}=2\\sqrt{15}$$", "input": "\\sqrt{60}", "steps": [ { "type": "interim", "title": "Prime factorization of $$60:{\\quad}2^{2}\\cdot\\:3\\cdot\\:5$$", "input": "60", "result": "=\\sqrt{2^{2}\\cdot\\:3\\cdot\\:5}", "steps": [ { "type": "step", "primary": "$$60\\:$$divides by $$2\\quad\\:60=30\\cdot\\:2$$", "result": "=2\\cdot\\:30" }, { "type": "step", "primary": "$$30\\:$$divides by $$2\\quad\\:30=15\\cdot\\:2$$", "result": "=2\\cdot\\:2\\cdot\\:15" }, { "type": "step", "primary": "$$15\\:$$divides by $$3\\quad\\:15=5\\cdot\\:3$$", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:5" }, { "type": "step", "primary": "$$2,\\:3,\\:5$$ are all prime numbers, therefore no further factorization is possible", "result": "=2\\cdot\\:2\\cdot\\:3\\cdot\\:5" }, { "type": "step", "result": "=2^{2}\\cdot\\:3\\cdot\\:5" } ], "meta": { "solvingClass": "Composite Integer", "interimType": "Prime Fac 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMrfn8NOj0LUzuzje6xTyxRuUHkFwKrCGUG/pR2kioRozwPjJo8Fi+uo29zEltAtWEB4gitN/2ICkrV6ivfiR3BLFRzd4QlsM8ugKm4vxBIECCPTamFM7zFum5hzgvbYsX" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{2^{2}}\\sqrt{3\\cdot\\:5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{2^{2}}=2$$" ], "result": "=2\\sqrt{3\\cdot\\:5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Refine", "result": "=2\\sqrt{15}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7g64l4Cz5BxdnpEUWG6uW1iAn9lkDfZkicUGkO3EF+IoGT+KSlWZuCyVPVwh+afrYAyJesRP4/SBgGpLgvEvz8QI0VkuLvmwx530aB0UhCWnASM2VLXBx9YxD7rfYrfSh" } }, { "type": "step", "result": "=\\frac{-6-2\\sqrt{15}i}{6}" }, { "type": "interim", "title": "Factor $$-6-i2\\sqrt{15}:{\\quad}-2\\left(3+i\\sqrt{15}\\right)$$", "input": "-6-i2\\sqrt{15}", "result": "=-\\frac{2\\left(3+i\\sqrt{15}\\right)}{6}", "steps": [ { "type": "step", "primary": "Rewrite as", "result": "=-2\\cdot\\:3-2i\\sqrt{15}" }, { "type": "step", "primary": "Factor out common term $$2$$", "result": "=-2\\left(3+i\\sqrt{15}\\right)", "meta": { "practiceLink": "/practice/factoring-practice", "practiceTopic": "Factoring" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "primary": "Cancel the common factor: $$2$$", "result": "=-\\frac{3+\\sqrt{15}i}{3}" }, { "type": "interim", "title": "Rewrite $$-\\frac{3+\\sqrt{15}i}{3}$$ in standard complex form: $$-1-\\sqrt{\\frac{5}{3}}i$$", "input": "-\\frac{3+\\sqrt{15}i}{3}", "steps": [ { "type": "step", "primary": "Apply the fraction rule: $$\\frac{a\\pm\\:b}{c}=\\frac{a}{c}\\pm\\:\\frac{b}{c}$$", "secondary": [ "$$\\frac{3+\\sqrt{15}i}{3}=-\\left(\\frac{3}{3}\\right)-\\left(\\frac{\\sqrt{15}i}{3}\\right)$$" ], "result": "=-\\left(\\frac{3}{3}\\right)-\\left(\\frac{\\sqrt{15}i}{3}\\right)" }, { "type": "step", "primary": "Remove parentheses: $$\\left(a\\right)=a$$", "result": "=-\\frac{3}{3}-\\frac{\\sqrt{15}i}{3}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=-1-\\frac{\\sqrt{15}i}{3}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{15}i}{3}:{\\quad}\\frac{\\sqrt{5}i}{\\sqrt{3}}$$", "input": "\\frac{\\sqrt{15}i}{3}", "steps": [ { "type": "interim", "title": "Factor $$\\sqrt{15}:{\\quad}\\sqrt{3}\\sqrt{5}$$", "steps": [ { "type": "step", "primary": "Factor $$15=3\\cdot\\:5$$", "result": "=\\sqrt{3\\cdot\\:5}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{ab}=\\sqrt[n]{a}\\sqrt[n]{b}$$", "result": "=\\sqrt{3}\\sqrt{5}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Algebraic Manipulation Factor Title 1Eq" } }, { "type": "step", "result": "=\\frac{\\sqrt{3}\\sqrt{5}i}{3}" }, { "type": "interim", "title": "Cancel $$\\frac{\\sqrt{3}\\sqrt{5}i}{3}:{\\quad}\\frac{\\sqrt{5}i}{\\sqrt{3}}$$", "input": "\\frac{\\sqrt{3}\\sqrt{5}i}{3}", "result": "=\\frac{\\sqrt{5}i}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a}=a^{\\frac{1}{n}}$$", "secondary": [ "$$\\sqrt{3}=3^{\\frac{1}{2}}$$" ], "result": "=\\frac{3^{\\frac{1}{2}}\\sqrt{5}i}{3}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } }, { "type": "step", "primary": "Apply exponent rule: $$\\frac{x^{a}}{x^{b}}=\\frac{1}{x^{b-a}}$$", "secondary": [ "$$\\frac{3^{\\frac{1}{2}}}{3^{1}}=\\frac{1}{3^{1-\\frac{1}{2}}}$$" ], "result": "=\\frac{\\sqrt{5}i}{3^{1-\\frac{1}{2}}}", "meta": { "practiceLink": "/practice/exponent-practice", "practiceTopic": "Expand FOIL" } }, { "type": "step", "primary": "Subtract the numbers: $$1-\\frac{1}{2}=\\frac{1}{2}$$", "result": "=\\frac{\\sqrt{5}i}{3^{\\frac{1}{2}}}" }, { "type": "step", "primary": "Apply radical rule: $$a^{\\frac{1}{n}}=\\sqrt[n]{a}$$", "secondary": [ "$$3^{\\frac{1}{2}}=\\sqrt{3}$$" ], "result": "=\\frac{\\sqrt{5}i}{\\sqrt{3}}", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYvrJHjtI0OUON9XwNWVjSFvkCWePEzHsqCEySB8gT/kAdYPfXQvX4/bINBB8wSEQ0ZOqgwlSBnSOLl0Z4MyJcDY0FcsgUXpB46zNLx8wApZ9WFYzcA4oaZR79v7JdxE/FtXbWAsR1oRbFHKj3hvINMkUoLp5qq1m+z9lyjAbATSwtJg9F0J8Eh8sNet1oT5qm78yD3hLQ33B7/8/LpbPE3o=" } } ], "meta": { "interimType": "Generic Cancel Title 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYkzVBiIB2gb2lMG3ELmwUP0AlilG71elit3w1IBbYN0PeH+rcvZk9LuAUQZeZZ0NvHrnyK4GJDrh3997WlBxCTQEsT0UExVGa239HC3hPbaceqXxdc+rps1CUyb7fqI2Gd0YWqP4RCoBmHFfiUsMXgOZXYapynkXr2nlBoJDcQfJvzIPeEtDfcHv/z8uls8Teg==" } }, { "type": "step", "result": "=-1-\\frac{\\sqrt{5}i}{\\sqrt{3}}" }, { "type": "interim", "title": "$$-\\frac{\\sqrt{5}}{\\sqrt{3}}=-\\sqrt{\\frac{5}{3}}$$", "input": "-\\frac{\\sqrt{5}}{\\sqrt{3}}", "steps": [ { "type": "step", "primary": "Combine same powers : $$\\frac{\\sqrt{x}}{\\sqrt{y}}=\\sqrt{\\frac{x}{y}}$$", "result": "=-\\sqrt{\\frac{5}{3}}" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7rpPPo+Nmcur7TLc75qZWRdaZCnUR2SjurESQtElIv3wtOtZYwUjyXhDTsNnn6ElrJZ7s7lv1pf9Tkz9FuXr4VSPNrQUrUslEDq5GoSnp4ZD/P/+v51eTuH2/F4MIu5mqafOiVtbdHS56L/gcvp8rPfQ3swCSwWDNToh16LWXsYlng1k1ogpUmBtxY0YLBR1lzru4mJyAykvNbli74dD+NA==" } }, { "type": "step", "result": "=-1-\\sqrt{\\frac{5}{3}}i" } ], "meta": { "interimType": "Rewrite In Complex Form Title 2Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74TqdRGDyv3oqElK9EevGSybeMBJT2c883KAPqIs+ngEWbJ8MPj+143aJFOCHJhoPfAu5u/TBlzVG5qXgF9PAhyjetd55DYlveZzsS8XHZnp6pfF1z6umzUJTJvt+ojYZwPTM+xnhoGkKkgIQ0VhzmCbeMBJT2c883KAPqIs+ngGT+MtufeumGM+GQY6Qiq9oQO9djmXGGOQuWkFSRSz08+WvT9lR6dzrjAd/DNDcBv0=" } }, { "type": "step", "result": "=-1-\\sqrt{\\frac{5}{3}}i" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7NsuMUt1MkxvwWUeZlTKnG8UkT6cnq/VummR4yg5fJ+Q+IPpCAKY4OB92iKXoehltdYPfXQvX4/bINBB8wSEQ0QzOnQozTOJXKKncWVdQGL3HMpwNPcPnS0bfrWSkcZZrRSpN33oxZMojoqvYhvSJACLkpiVA7S8UT8ieezZKu6pDtdnxXXyBnscDa1cW1xp1RVlcYKwWmYW0y9dYRJhTMTX6Fk5J9NjmVWcOuIHp624=" } }, { "type": "step", "result": "x=-1-i\\sqrt{\\frac{5}{3}}" } ], "meta": { "solvingClass": "Equations", "practiceLink": "/practice/linear-equations-practice", "practiceTopic": "Linear Equations" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x=\\frac{-6-\\sqrt{6^{2}-96}}{2\\cdot 3}" }, "showViewLarger": true } }, "meta": { "showVerify": true } }