{
"query": {
"display": "intercepts $$f\\left(x\\right)=2x^{3}+6x^{2}-90x+5$$",
"symbolab_question": "CONIC#intercepts f(x)=2x^{3}+6x^{2}-90x+5"
},
"solution": {
"level": "PERFORMED",
"subject": "Functions & Graphing",
"topic": "Functions",
"subTopic": "intercepts",
"default": "\\mathrm{X\\:Intercepts}: (0.05576…,0),(5.33972…,0),(-8.39548…,0),\\mathrm{Y\\:Intercepts}: (0,5)",
"meta": {
"showVerify": true
}
},
"steps": {
"type": "interim",
"title": "Axis interception points of $$2x^{3}+6x^{2}-90x+5:\\quad\\:$$X Intercepts$$:\\:\\left(0.05576…,\\:0\\right),\\:\\left(5.33972…,\\:0\\right),\\:\\left(-8.39548…,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:5\\right)$$",
"steps": [
{
"type": "interim",
"title": "$$x-$$axis interception points of $$2x^{3}+6x^{2}-90x+5:{\\quad}\\left(0.05576…,\\:0\\right),\\:\\left(5.33972…,\\:0\\right),\\:\\left(-8.39548…,\\:0\\right)$$",
"input": "2x^{3}+6x^{2}-90x+5",
"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 $$2x^{3}+6x^{2}-90x+5=0:{\\quad}x\\approx\\:0.05576…,\\:x\\approx\\:5.33972…,\\:x\\approx\\:-8.39548…$$",
"input": "2x^{3}+6x^{2}-90x+5=0",
"steps": [
{
"type": "interim",
"title": "Find one solution for $$2x^{3}+6x^{2}-90x+5=0$$ using Newton-Raphson:$${\\quad}x\\approx\\:0.05576…$$",
"input": "2x^{3}+6x^{2}-90x+5=0",
"steps": [
{
"type": "definition",
"title": "Newton-Raphson Approximation Definition",
"text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$"
},
{
"type": "step",
"result": "f\\left(x\\right)=2x^{3}+6x^{2}-90x+5"
},
{
"type": "interim",
"title": "Find $$f^{^{\\prime}}\\left(x\\right):{\\quad}6x^{2}+12x-90$$",
"input": "\\frac{d}{dx}\\left(2x^{3}+6x^{2}-90x+5\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{d}{dx}\\left(2x^{3}\\right)+\\frac{d}{dx}\\left(6x^{2}\\right)-\\frac{d}{dx}\\left(90x\\right)+\\frac{d}{dx}\\left(5\\right)"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2x^{3}\\right)=6x^{2}$$",
"input": "\\frac{d}{dx}\\left(2x^{3}\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{d}{dx}\\left(x^{3}\\right)"
},
{
"type": "step",
"primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=2\\cdot\\:3x^{3-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "step",
"primary": "Simplify",
"result": "=6x^{2}",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnw/L9p+IMyouNhbzU7bNo6TdaV09PMxEKZ9FieghTFwWn+xgzF413/kfGYgWrraBxHO0oTnnZveyzJ4AtC1ZGNjDT5Dj/fM73/u0bafjbUvQ9geJFbKg/ol2YLcmIH9YSS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(6x^{2}\\right)=12x$$",
"input": "\\frac{d}{dx}\\left(6x^{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=6\\frac{d}{dx}\\left(x^{2}\\right)"
},
{
"type": "step",
"primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=6\\cdot\\:2x^{2-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "step",
"primary": "Simplify",
"result": "=12x",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYk9DkJJyTtuysYPqODNgX1aTdaV09PMxEKZ9FieghTFwrRUIuzIwH26WWqbPQjOKWGRLd2VwIqlBNByF6663syQKWDcfYGU6NsijW8FXhyvwDoFlJZuQZisuobnBchqNSQ=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(90x\\right)=90$$",
"input": "\\frac{d}{dx}\\left(90x\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=90\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$",
"result": "=90\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=90",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYnjbjl7ART4c1WIULhbtpoiXIQHgliMhSOSNsNni19In4XKqwbRRdso83xDz8lW4+w4bfwiV6iMLJ5sC1nL7dOasX6FQwg66N5WdYhvIgtzLsIjaxJ4DvjTb2fbKjbvtlQ=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(5\\right)=0$$",
"input": "\\frac{d}{dx}\\left(5\\right)",
"steps": [
{
"type": "step",
"primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYmXEh6/dOKVl5+UiJ6t4qwxJ8Vk6wvKjVnTtwWT18bQnz7FeFrf3rcM8IZlDz2c0dm5O2bEw0Ql6ne7k1AUriTvz/OzRy6l5fd6++0L3aMbw"
}
},
{
"type": "step",
"result": "=6x^{2}+12x-90+0"
},
{
"type": "step",
"primary": "Simplify",
"result": "=6x^{2}+12x-90",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Generic Find Title 1Eq"
}
},
{
"type": "step",
"primary": "Let $$x_{0}=0$$",
"secondary": [
"Compute $$x_{n+1}$$ until $$\\Delta\\:x_{n+1}\\:<\\:0.000001$$"
]
},
{
"type": "interim",
"title": "$$x_{1}=0.05555…{\\quad:\\quad}Δx_{1}=0.05555…$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{0}\\right)=2\\cdot\\:0^{3}+6\\cdot\\:0^{2}-90\\cdot\\:0+5=5$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{0}\\right)=6\\cdot\\:0^{2}+12\\cdot\\:0-90=-90$$",
"$$x_{1}=0-\\frac{5}{-90}=0.05555…$$"
],
"result": "x_{1}=0.05555…"
},
{
"type": "step",
"primary": "$$Δx_{1}=\\left|0.05555…-0\\right|=0.05555…$$",
"result": "Δx_{1}=0.05555…"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "$$x_{2}=0.05576…{\\quad:\\quad}Δx_{2}=0.00021…$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{1}\\right)=2\\cdot\\:0.05555…^{3}+6\\cdot\\:0.05555…^{2}-90\\cdot\\:0.05555…+5=0.01886…$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{1}\\right)=6\\cdot\\:0.05555…^{2}+12\\cdot\\:0.05555…-90=-89.31481…$$",
"$$x_{2}=0.05555…-\\frac{0.01886…}{-89.31481…}=0.05576…$$"
],
"result": "x_{2}=0.05576…"
},
{
"type": "step",
"primary": "$$Δx_{2}=\\left|0.05576…-0.05555…\\right|=0.00021…$$",
"result": "Δx_{2}=0.00021…"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "$$x_{3}=0.05576…{\\quad:\\quad}Δx_{3}=3.16267E-9$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{2}\\right)=2\\cdot\\:0.05576…^{3}+6\\cdot\\:0.05576…^{2}-90\\cdot\\:0.05576…+5=2.82465E-7$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{2}\\right)=6\\cdot\\:0.05576…^{2}+12\\cdot\\:0.05576…-90=-89.31213…$$",
"$$x_{3}=0.05576…-\\frac{2.82465E-7}{-89.31213…}=0.05576…$$"
],
"result": "x_{3}=0.05576…"
},
{
"type": "step",
"primary": "$$Δx_{3}=\\left|0.05576…-0.05576…\\right|=3.16267E-9$$",
"result": "Δx_{3}=3.16267E-9"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "x\\approx\\:0.05576…"
}
],
"meta": {
"interimType": "Newton Raphson Find Real Solution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjD0Q+Kmlzmk3HAaHUr0qBgwHZ6HKXjyUOxOBEQ1QUIHB69OOsSYPyjmaBpxiizP0AaA2DPiSvwI+Z8VwydOBgDmfKgwS/hc2MJbAPRfUaxt2ng45XG7mtI3pmDdmjXsOLuH63v9QR0jPDNEngIk7RBiTeQKHeh69S6dnv9vSoUoFNOWDEPsOuhFfbj166v7v+Hc246TvMvSwJMYNDpbh3VCtpbpmiKbpW/hJ6BLayz/baBHI1r1uQU4GmawA8goVAf7VQpU9yh+VO0khWxtpXKq"
}
},
{
"type": "step",
"primary": "Apply long division:$${\\quad}\\frac{2x^{3}+6x^{2}-90x+5}{x-0.05576…}=2x^{2}+6.11153…x-89.65917…$$"
},
{
"type": "step",
"result": "2x^{2}+6.11153…x-89.65917…\\approx\\:0"
},
{
"type": "interim",
"title": "Find one solution for $$2x^{2}+6.11153…x-89.65917…=0$$ using Newton-Raphson:$${\\quad}x\\approx\\:5.33972…$$",
"input": "2x^{2}+6.11153…x-89.65917…=0",
"steps": [
{
"type": "definition",
"title": "Newton-Raphson Approximation Definition",
"text": "The Newton-Raphson method uses an iterative process to approach one root of a function<br/>$$x_{n+1}=x_{n}\\:-\\:\\frac{f\\left(x_{n}\\right)}{f^{\\prime}\\left(x_{n}\\right)}$$"
},
{
"type": "step",
"result": "f\\left(x\\right)=2x^{2}+6.11153…x-89.65917…"
},
{
"type": "interim",
"title": "Find $$f^{^{\\prime}}\\left(x\\right):{\\quad}4x+6.11153…$$",
"input": "\\frac{d}{dx}\\left(2x^{2}+6.11153…x-89.65917…\\right)",
"steps": [
{
"type": "step",
"primary": "Apply the Sum/Difference Rule: $$\\left(f{\\pm}g\\right)'=f'{\\pm}g'$$",
"result": "=\\frac{d}{dx}\\left(2x^{2}\\right)+\\frac{d}{dx}\\left(6.11153…x\\right)-\\frac{d}{dx}\\left(89.65917…\\right)"
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(2x^{2}\\right)=4x$$",
"input": "\\frac{d}{dx}\\left(2x^{2}\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=2\\frac{d}{dx}\\left(x^{2}\\right)"
},
{
"type": "step",
"primary": "Apply the Power Rule: $$\\frac{d}{dx}\\left(x^a\\right)=a{\\cdot}x^{a-1}$$",
"result": "=2\\cdot\\:2x^{2-1}",
"meta": {
"practiceLink": "/practice/derivatives-practice#area=main&subtopic=Power%20Rule",
"practiceTopic": "Power Rule"
}
},
{
"type": "step",
"primary": "Simplify",
"result": "=4x",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYgEShBvy1snibnAN3NvB1a2TdaV09PMxEKZ9FieghTFwHBO3D9VaGp1eOVvjTiCiEaN6Hv6MoTMtvtU0IQwXdn+XNwOQ43NHE8cpERrPgoqpfTZuddhTh3r/FmyVu4x1Bw=="
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(6.11153…x\\right)=6.11153…$$",
"input": "\\frac{d}{dx}\\left(6.11153…x\\right)",
"steps": [
{
"type": "step",
"primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$",
"result": "=6.11153…\\frac{dx}{dx}"
},
{
"type": "step",
"primary": "Apply the common derivative: $$\\frac{dx}{dx}=1$$",
"result": "=6.11153…\\cdot\\:1"
},
{
"type": "step",
"primary": "Simplify",
"result": "=6.11153…",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYn7W7UEFe9zGPiFBv3xE2EBs3bzEbqKu8+A9drXdKSFLqKwXnDyHJSOk7SW/uMHpmB6N8WgZmiYU3p71nm4L2qgOG38IleojCyebAtZy+3TmTt6L/qF2LGvLz28K1znYUl3UcixnYxlGwz7rwZ8LZNewiNrEngO+NNvZ9sqNu+2V"
}
},
{
"type": "interim",
"title": "$$\\frac{d}{dx}\\left(89.65917…\\right)=0$$",
"input": "\\frac{d}{dx}\\left(89.65917…\\right)",
"steps": [
{
"type": "step",
"primary": "Derivative of a constant: $$\\frac{d}{dx}\\left({a}\\right)=0$$",
"result": "=0"
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Derivatives",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYg2IuOkFvfOKWEIp+PM5Kv9KSCY1fqKE9KQuJfhrlGtxqKwXnDyHJSOk7SW/uMHpmILlWngNL5BbQEd4M6iPSGdjDT5Dj/fM73/u0bafjbUvwPp5ibnHWJcmJ13qYLo6ZyS3daIZHtloJpe/PvtsyNI="
}
},
{
"type": "step",
"result": "=4x+6.11153…-0"
},
{
"type": "step",
"primary": "Simplify",
"result": "=4x+6.11153…",
"meta": {
"solvingClass": "Solver"
}
}
],
"meta": {
"solvingClass": "Derivatives",
"interimType": "Generic Find Title 1Eq"
}
},
{
"type": "step",
"primary": "Let $$x_{0}=5$$",
"secondary": [
"Compute $$x_{n+1}$$ until $$\\Delta\\:x_{n+1}\\:<\\:0.000001$$"
]
},
{
"type": "interim",
"title": "$$x_{1}=5.34856…{\\quad:\\quad}Δx_{1}=0.34856…$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{0}\\right)=2\\cdot\\:5^{2}+6.11153…\\cdot\\:5-89.65917…=-9.10151…$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{0}\\right)=4\\cdot\\:5+6.11153…=26.11153…$$",
"$$x_{1}=5-\\frac{-9.10151…}{26.11153…}=5.34856…$$"
],
"result": "x_{1}=5.34856…"
},
{
"type": "step",
"primary": "$$Δx_{1}=\\left|5.34856…-5\\right|=0.34856…$$",
"result": "Δx_{1}=0.34856…"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "$$x_{2}=5.33972…{\\quad:\\quad}Δx_{2}=0.00883…$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{1}\\right)=2\\cdot\\:5.34856…^{2}+6.11153…\\cdot\\:5.34856…-89.65917…=0.24299…$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{1}\\right)=4\\cdot\\:5.34856…+6.11153…=27.50578…$$",
"$$x_{2}=5.34856…-\\frac{0.24299…}{27.50578…}=5.33972…$$"
],
"result": "x_{2}=5.33972…"
},
{
"type": "step",
"primary": "$$Δx_{2}=\\left|5.33972…-5.34856…\\right|=0.00883…$$",
"result": "Δx_{2}=0.00883…"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "$$x_{3}=5.33972…{\\quad:\\quad}Δx_{3}=5.682E-6$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{2}\\right)=2\\cdot\\:5.33972…^{2}+6.11153…\\cdot\\:5.33972…-89.65917…=0.00015…$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{2}\\right)=4\\cdot\\:5.33972…+6.11153…=27.47044…$$",
"$$x_{3}=5.33972…-\\frac{0.00015…}{27.47044…}=5.33972…$$"
],
"result": "x_{3}=5.33972…"
},
{
"type": "step",
"primary": "$$Δx_{3}=\\left|5.33972…-5.33972…\\right|=5.682E-6$$",
"result": "Δx_{3}=5.682E-6"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "interim",
"title": "$$x_{4}=5.33972…{\\quad:\\quad}Δx_{4}=2.34964E-12$$",
"steps": [
{
"type": "step",
"primary": "$$f\\left(x_{3}\\right)=2\\cdot\\:5.33972…^{2}+6.11153…\\cdot\\:5.33972…-89.65917…=6.45457E-11$$",
"secondary": [
"$$f^{^{\\prime}}\\left(x_{3}\\right)=4\\cdot\\:5.33972…+6.11153…=27.47042…$$",
"$$x_{4}=5.33972…-\\frac{6.45457E-11}{27.47042…}=5.33972…$$"
],
"result": "x_{4}=5.33972…"
},
{
"type": "step",
"primary": "$$Δx_{4}=\\left|5.33972…-5.33972…\\right|=2.34964E-12$$",
"result": "Δx_{4}=2.34964E-12"
}
],
"meta": {
"interimType": "N/A"
}
},
{
"type": "step",
"result": "x\\approx\\:5.33972…"
}
],
"meta": {
"interimType": "Newton Raphson Find Real Solution 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjD0Q+Kmlzmk3HAaHUr0qBgwHu9ebHVtrlhBlrUlSv5IiaKUPveyB+7On9kitV3JfLUtmzlZ7ns5ZS6RsP0ADq//OIkUwKTUismV87U9kJNpMF7qYI8hViGSsSBJMsQ7M3E/dcOwHFHHSxX5Kpjk7vVO072c4dO4LP7EmSE61O9pPHimqO4VtL6M91XE4IPuJJtOo4pfCvNjInl9LE2Pd8LH6nNoCBWpfv/v3czM2EJvKIKmOfx7+Nxb0c2SBfRun5/UjBYZvtBxC7U3Aqr7nfs6ueQU/zSzS16yyMUJvFtrg4mpXFf3SOUx+H18qfp3MLg="
}
},
{
"type": "step",
"primary": "Apply long division:$${\\quad}\\frac{2x^{2}+6.11153…x-89.65917…}{x-5.33972…}=2x+16.79097…$$"
},
{
"type": "step",
"result": "2x+16.79097…\\approx\\:0"
},
{
"type": "step",
"result": "x\\approx\\:-8.39548…"
},
{
"type": "step",
"primary": "The solutions are",
"result": "x\\approx\\:0.05576…,\\:x\\approx\\:5.33972…,\\:x\\approx\\:-8.39548…"
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "\\left(0.05576…,\\:0\\right),\\:\\left(5.33972…,\\:0\\right),\\:\\left(-8.39548…,\\:0\\right)"
}
],
"meta": {
"solvingClass": "Function Intersect",
"interimType": "Interception X Points Top 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7P3fQzVTF/QWWMG/vdjSvc856ljGNNrOfXFSxgyE42paAXx8aiPEJP8/Af8Qp/iCNIZ9T38S5pPc5mtR18r12ZvXqifj1ZU6pvAftTJBX5HZ1OQ7KEPbFtFozyuQjXbc+gkt3WiGR7ZaCaXvz77bMjS"
}
},
{
"type": "interim",
"title": "$$y-$$axis interception point of $$2x^{3}+6x^{2}-90x+5:{\\quad}\\left(0,\\:5\\right)$$",
"input": "2x^{3}+6x^{2}-90x+5",
"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 $$y=2\\cdot\\:0^{3}+6\\cdot\\:0^{2}-90\\cdot\\:0+5:{\\quad}y=5$$",
"input": "y=2\\cdot\\:0^{3}+6\\cdot\\:0^{2}-90\\cdot\\:0+5",
"steps": [
{
"type": "interim",
"title": "Simplify $$2\\cdot\\:0^{3}+6\\cdot\\:0^{2}-90\\cdot\\:0+5:{\\quad}5$$",
"input": "2\\cdot\\:0^{3}+6\\cdot\\:0^{2}-90\\cdot\\:0+5",
"steps": [
{
"type": "step",
"primary": "Apply rule $$0^{a}=0$$",
"secondary": [
"$$0^{3}=0,\\:0^{2}=0$$"
],
"result": "=2\\cdot\\:0+6\\cdot\\:0-90\\cdot\\:0+5"
},
{
"type": "step",
"primary": "Apply rule $$0\\cdot\\:a=0$$",
"result": "=0+0-0+5"
},
{
"type": "step",
"primary": "Add/Subtract the numbers: $$0+0-0+5=5$$",
"result": "=5"
}
],
"meta": {
"solvingClass": "Solver",
"interimType": "Generic Simplify Specific 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7ZGP46nDK9G4QtUcdibLhHAqnjz6KN1/fiOXQvpgcxO07h/7JoEleqokOWxjoEIlTzRqDxPUzBN6vjj5oJL9kUJVnNc0dafeBGgJCXbcaKBV6pfF1z6umzUJTJvt+ojYZa+XIqT7+sGIIONXYNLACJN8aYH95G+LOnkqN2MQkg7Vb3TydgomyuSUwWnsdNdR1X2j5dL49Un/qqDXI2U1chw=="
}
},
{
"type": "step",
"result": "y=5"
}
],
"meta": {
"solvingClass": "Equations",
"interimType": "Generic Solve Title 1Eq"
}
},
{
"type": "step",
"result": "\\left(0,\\:5\\right)"
}
],
"meta": {
"solvingClass": "Function Intersect",
"interimType": "Interception Y Points Top 1Eq",
"gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcflHbOhz+MjaigESSQwPfHkrR9w8veN2Fgw2WihY6OBcbWHQN8EyYXYnu0ZMK7ir7WlpD04iGaHK6DGeW80sBZzrvwo8YjJdATZbHJ+k/9klGzgT2ATl+Uc9GfrN+14OPTg="
}
},
{
"type": "step",
"result": "\\mathrm{X\\:Intercepts}:\\:\\left(0.05576…,\\:0\\right),\\:\\left(5.33972…,\\:0\\right),\\:\\left(-8.39548…,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:5\\right)"
}
],
"meta": {
"solvingClass": "Function Intersect"
}
},
"plot_output": {
"meta": {
"plotInfo": {
"variable": "x",
"plotRequest": "2x^{3}+6x^{2}-90x+5"
},
"showViewLarger": true
}
},
"meta": {
"showVerify": true
}
}
Solution
intercepts
Solution
Solution steps
axis interception points of
axis interception point of