intercepts of f(x)=x^2+2x-3

{ "query": { "display": "intercepts $$f\\left(x\\right)=x^{2}+2x-3$$", "symbolab_question": "CONIC#intercepts f(x)=x^{2}+2x-3" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "intercepts", "default": "\\mathrm{X\\:Intercepts}: (1,0),(-3,0),\\mathrm{Y\\:Intercepts}: (0,-3)", "meta": { "showVerify": true } }, "steps": { "type": "interim", "title": "Axis interception points of $$x^{2}+2x-3:\\quad\\:$$X Intercepts$$:\\:\\left(1,\\:0\\right),\\:\\left(-3,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:-3\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$x^{2}+2x-3:{\\quad}\\left(1,\\:0\\right),\\:\\left(-3,\\:0\\right)$$", "input": "x^{2}+2x-3", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "Solve $$x^{2}+2x-3=0:{\\quad}x=1,\\:x=-3$$", "input": "x^{2}+2x-3=0", "steps": [ { "type": "interim", "title": "Solve with the quadratic formula", "input": "x^{2}+2x-3=0", "result": "{x}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(-3\\right)}}{2\\cdot\\:1}", "steps": [ { "type": "definition", "title": "Quadratic Equation Formula:", "text": "For a quadratic equation of the form $$ax^2+bx+c=0$$ the solutions are <br/>$${\\quad}x_{1,\\:2}=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}$$" }, { "type": "step", "primary": "For $${\\quad}a=1,\\:b=2,\\:c=-3$$", "result": "{x}_{1,\\:2}=\\frac{-2\\pm\\:\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(-3\\right)}}{2\\cdot\\:1}" } ], "meta": { "interimType": "Solving The Quadratic Equation With Quadratic Formula Definition 0Eq", "gptData": "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" } }, { "type": "interim", "title": "$$\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(-3\\right)}=4$$", "input": "\\sqrt{2^{2}-4\\cdot\\:1\\cdot\\:\\left(-3\\right)}", "result": "{x}_{1,\\:2}=\\frac{-2\\pm\\:4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Apply rule $$-\\left(-a\\right)=a$$", "result": "=\\sqrt{2^{2}+4\\cdot\\:1\\cdot\\:3}" }, { "type": "step", "primary": "Multiply the numbers: $$4\\cdot\\:1\\cdot\\:3=12$$", "result": "=\\sqrt{2^{2}+12}" }, { "type": "step", "primary": "$$2^{2}=4$$", "result": "=\\sqrt{4+12}" }, { "type": "step", "primary": "Add the numbers: $$4+12=16$$", "result": "=\\sqrt{16}" }, { "type": "step", "primary": "Factor the number: $$16=4^{2}$$", "result": "=\\sqrt{4^{2}}" }, { "type": "step", "primary": "Apply radical rule: $$\\sqrt[n]{a^n}=a$$", "secondary": [ "$$\\sqrt{4^{2}}=4$$" ], "result": "=4", "meta": { "practiceLink": "/practice/radicals-practice", "practiceTopic": "Radical Rules" } } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s74YGLq7ZDKNPpn08HXk5ZmzTF392od8JjfT80CJ7FSjmzNcis0Fl12BUtPUACi5rezMFYmi1F5Hg/ibpEToVnYz8s1oAoBG9v5706TVT42P68crMamW+jharfA0DCQlleNMXf3ah3wmN9PzQInsVKOY85tdVMH0xtZw1ah+Gjip8=" } }, { "type": "step", "primary": "Separate the solutions", "result": "{x}_{1}=\\frac{-2+4}{2\\cdot\\:1},\\:{x}_{2}=\\frac{-2-4}{2\\cdot\\:1}" }, { "type": "interim", "title": "$$x=\\frac{-2+4}{2\\cdot\\:1}:{\\quad}1$$", "input": "\\frac{-2+4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Add/Subtract the numbers: $$-2+4=2$$", "result": "=\\frac{2}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{2}{2}" }, { "type": "step", "primary": "Apply rule $$\\frac{a}{a}=1$$", "result": "=1" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7CtPEPwLR6ThZzZnyAQrG5CJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5l1K1r0aaVT/E0zZakID9jPAKCfFHLER2iRFCuUUu6VNUlZ0WfCB+stvMQJpka/avsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "$$x=\\frac{-2-4}{2\\cdot\\:1}:{\\quad}-3$$", "input": "\\frac{-2-4}{2\\cdot\\:1}", "steps": [ { "type": "step", "primary": "Subtract the numbers: $$-2-4=-6$$", "result": "=\\frac{-6}{2\\cdot\\:1}" }, { "type": "step", "primary": "Multiply the numbers: $$2\\cdot\\:1=2$$", "result": "=\\frac{-6}{2}" }, { "type": "step", "primary": "Apply the fraction rule: $$\\frac{-a}{b}=-\\frac{a}{b}$$", "result": "=-\\frac{6}{2}" }, { "type": "step", "primary": "Divide the numbers: $$\\frac{6}{2}=3$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Solver", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7I0WjRHGFt5FzboxdnuoLuSJzNSds22HF6C2iiwwKPXMDnzlbPZjyKgy1eUCFsLd5QW9qS4M5U2Tj2CY8RpIMUL5EbOPmFIVDZCKr5u8wDXo/VIK3pNXNJkrd9QpHfgzoialcV/dI5TH4fXyp+ncwuA==" } }, { "type": "step", "primary": "The solutions to the quadratic equation are:", "result": "x=1,\\:x=-3" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(1,\\:0\\right),\\:\\left(-3,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7NJ8u2HESF7lrMZQOrRsgM4LAY1Tpv5VpOWP6f3TmJ0julcBNNeXoGI0NE3bKqzuby7eYPigeHu2+othUfpPT+zscOczUJPLP7Ke/HI544azg==" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$x^{2}+2x-3:{\\quad}\\left(0,\\:-3\\right)$$", "input": "x^{2}+2x-3", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=0^{2}+2\\cdot\\:0-3:{\\quad}y=-3$$", "input": "y=0^{2}+2\\cdot\\:0-3", "steps": [ { "type": "interim", "title": "Simplify $$0^{2}+2\\cdot\\:0-3:{\\quad}-3$$", "input": "0^{2}+2\\cdot\\:0-3", "steps": [ { "type": "step", "primary": "Apply rule $$0^{a}=0$$", "secondary": [ "$$0^{2}=0$$" ], "result": "=0+2\\cdot\\:0-3" }, { "type": "step", "primary": "Apply rule $$0\\cdot\\:a=0$$", "result": "=0+0-3" }, { "type": "step", "primary": "Add/Subtract the numbers: $$0+0-3=-3$$", "result": "=-3" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7LsAGgKAuFT6BaxF1EpDQYtk18MgjmsLVrysQjUr/8hGjkVi15I8rBefLi4Iyt2wrajlVg7zaIiVTSziPPJVJkx4pgUWEah0lniZLlD4X0wucB7w92RzcTJdZZ7ZO46NrQGBC0Mbxe7q5kCCu1Xc+Wg==" } }, { "type": "step", "result": "y=-3" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:-3\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfkVDHoJxzud7B4H/6p0HynqasQjB7dlbvUVQiIaNBx6s6GgBg7+Qqjt3KiwKdNp2BIKKOeNRJWkuPA4seA3gb5zo5KAbAqPuzqL2wFb6i4zJg==" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(1,\\:0\\right),\\:\\left(-3,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:-3\\right)" } ], "meta": { "solvingClass": "Function Intersect" } }, "plot_output": { "meta": { "plotInfo": { "variable": "x", "plotRequest": "x^{2}+2x-3" }, "showViewLarger": true } }, "meta": { "showVerify": true } }