MATH MADE EASY - MathMaster

=x3-2x2+3x-1+4x2-x+2","","Solution: \u003Cbr />\n1. Write down the given polynomials:\u003Cbr />\n- First polynomial: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>g

$x$ = x^3 - 2x^2 + 3x - 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- Second polynomial: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x^2 - x + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Align and add the polynomials term by term:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>g

$x$ = x^3 - 2x^2 + 3x - 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x^2 - x + 2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Add the corresponding like terms:\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^2\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-2x^2 + 4x^2 = 2x^2\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field> terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x - x = 2x\u003C/math-field>\u003C/math-field>\u003Cbr />\n- For constant terms: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>-1 + 2 = 1\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. The resulting polynomial after addition is:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x^3 + 2x^2 + 2x + 1\u003C/math-field>\u003C/math-field>",739,148,"add-the-polynomials-g-x-x3-2x2-3x-1-4x2-x-2",{"id":53,"category":45,"text_question":54,"photo_question":47,"text_answer":55,"step_text_answer":8,"step_photo_answer":8,"views":56,"likes":57,"slug":58},537912,"show how to do 14+231 in a base 12 system","Solution:\u003Cbr />\n1. Express the numbers in their base 10 equivalent:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} = 1 \\cdot 12^1 + 4 \\cdot 12^0 = 12 + 4 = 16\u003C/math-field>\u003C/math-field>\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>231_{12} = 2 \\cdot 12^2 + 3 \\cdot 12^1 + 1 \\cdot 12^0 = 288 + 36 + 1 = 325\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. Add the numbers in base 10:\u003Cbr />\n- \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>16 + 325 = 341\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n3. Convert the sum back to base 12:\u003Cbr />\n- To convert 341 to base 12, divide by 12 and keep track of the remainders.\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>341 \\div 12 = 28\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5\u003C/math-field>\u003C/math-field>

$the rightmost digit$ \u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>28 \\div 12 = 2\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4\u003C/math-field>\u003C/math-field>\u003Cbr />\n - \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2 \\div 12 = 0\u003C/math-field>\u003C/math-field> with remainder \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>2\u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. Read the remainders from last to first to get \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>245_{12}\u003C/math-field>\u003C/math-field>.\u003Cbr />\n\u003Cbr />\nThus, \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>14_{12} + 231_{12} = 245_{12}\u003C/math-field>\u003C/math-field>.",1182,236,"show-how-to-do-14-231-in-a-base-12-system",{"id":60,"category":45,"text_question":61,"photo_question":47,"text_answer":62,"step_text_answer":8,"step_photo_answer":8,"views":63,"likes":64,"slug":65},537889,"Two angels are supplementary if the sun of their angels is 180 degrees. One of the two supplementary angles is 8 degrees less than three times the other. Find the measure of each angle.","**\u003Cbr />\n\u003Cbr />\n1. Define the angles: Let the first angle be \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>. The second angle is \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x - 8\u003C/math-field>\u003C/math-field>.\u003Cbr />\n2. Set up the equation for supplementary angles: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x +

$3x - 8$ = 180\u003C/math-field>\u003C/math-field>.\u003Cbr />\n3. Simplify the equation: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x - 8 = 180\u003C/math-field>\u003C/math-field>.\u003Cbr />\n4. Solve for \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>: \u003Cbr />\n - Add 8 to both sides: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>4x = 188\u003C/math-field>\u003C/math-field>.\u003Cbr />\n - Divide by 4: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 47\u003C/math-field>\u003C/math-field>.\u003Cbr />\n5. Find the second angle: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>3x - 8 = 3

$47$ - 8 = 141 - 8 = 133\u003C/math-field>\u003C/math-field>.\u003Cbr />\n6. Conclusion: The angles are 47 degrees and 133 degrees.",1361,272,"two-angels-are-supplementary-if-the-sun-of-their-angels-is-180-degrees-one-of-the-two-supplementary-angles-is-8-degrees-less-than-three-times-the-other-find-the-measure-of-each-angle",{"id":67,"category":45,"text_question":68,"photo_question":47,"text_answer":69,"step_text_answer":8,"step_photo_answer":8,"views":70,"likes":71,"slug":72},537802,"1/2 x 2 1/2=","1. Convert the mixed number to an improper fraction: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\$2\\frac{1}{2} = \\frac{5}{2}\$\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>2. Multiply: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\$\\frac{1}{2} \\times \\frac{5}{2} = \\frac{1 \\times 5}{2 \\times 2} = \\frac{5}{4}\$\u003C/math-field>\u003C/math-field> .\u003Cbr>\u003Cbr>3. The answer is: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\$\\frac{5}{4}\$\u003C/math-field>\u003C/math-field> . \u003Cdiv>or in mixed number: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>1\\frac{1}{4}\u003C/math-field>\u003C/math-field>\u003C/div>",1064,213,"1-2-x-2-1-2",{"id":74,"category":45,"text_question":75,"photo_question":47,"text_answer":76,"step_text_answer":8,"step_photo_answer":8,"views":77,"likes":78,"slug":79},537741,"f(x)=x²\nF(g)=5x-2\nf°g","1. Define the functions: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f(x) = x^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> g(x) = 5x - 2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n2. The composition f°g means f(g(x)), which is plugging g(x) into f(x).\u003Cbr />\n\u003Cbr />\n3. Substitute g(x) into f(x): \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f(g(x)) = f(5x - 2) \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n4. This becomes: \u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> (5x - 2)^2 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\n5. Expand the square:\u003Cbr />\n \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> 25x^2 - 20x + 4 \u003C/math-field>\u003C/math-field>\u003Cbr />\n\u003Cbr />\nAnswer:\u003Cbr />\n\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only> f(g(x)) = 25x^2 - 20x + 4 \u003C/math-field>\u003C/math-field>",376,75,"f-x-x-f-g-5x-2-f-g",{"id":81,"category":45,"text_question":82,"photo_question":47,"text_answer":83,"step_text_answer":8,"step_photo_answer":8,"views":84,"likes":85,"slug":86},537652,"In a company, the accounting structure is based on the double-entry principle and consists of:\n a) Resources available to achieve the objectives set as a goal\n b) The sources of these, which are also indicative of the various liabilities incurred.\n Provide an example where this principle is demonstrated, clearly identifying resources and sources.","El principio de partida doble establece que cada transacción en la contabilidad afecta al menos a dos cuentas: una cuenta se debita y otra se acredita, manteniendo así el equilibrio contable.\n\nEjemplo Práctico\nTransacción: La empresa \"XYZ S.A.\" compra un equipo de cómputo por un valor de $1,000, pagando en efectivo.\n\nRecursos (Activos)\nEquipo de cómputo: $1,000 (Activo)\nFuentes (Pasivos y Patrimonio)\nEfectivo: -$1,000 (Disminución en el Activo Circulante)\nRegistro Contable\nDébito (Aumento en Activo):\nEquipo de cómputo: $1,000\nCrédito (Disminución en Activo):\nEfectivo: $1,000\nBalance\nActivos Totales:\nEquipo de cómputo: $1,000\nEfectivo: $0 (después de la compra)\nPasivos y Patrimonio:\nPasivos: $0\nPatrimonio: $0 (sin cambios en este caso)\nEste ejemplo demuestra cómo los recursos (el equipo de cómputo) y las fuentes (disminución de efectivo) se afectan mutuamente, cumpliendo con el principio de partida doble y asegurando que la contabilidad permanezca equilibrada.",392,78,"in-a-company-the-accounting-structure-is-based-on-the-double-entry-principle-and-consists-of-a-resources-available-to-achieve-the-objectives-set-as-a-goal-b-the-sources-of-these-which-are-also",{"id":88,"category":45,"text_question":89,"photo_question":47,"text_answer":90,"step_text_answer":8,"step_photo_answer":8,"views":91,"likes":92,"slug":93},537624,"They sold 13200 students tickets and 10450 for the next game. A stadium sold 12340 students tickets and 15670 general tickets for the first game. How many more tickets were sold for the first game? General tickets.","1. Calculate the total number of general tickets sold for the first game: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>15670\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>2. Calculate the total number of general tickets sold for the next game: \u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>10450\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>3. Find the difference in general tickets sold between the two games:\u003Cbr>\u003Cbr>\u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>15670-10450=5220\u003C/math-field>\u003C/math-field> \u003Cbr>\u003Cbr>Therefore, more 5220 general tickets were sold for the first game.",890,178,"they-sold-13200-students-tickets-and-10450-for-the-next-game-a-stadium-sold-12340-students-tickets-and-15670-general-tickets-for-the-first-game-how-many-more-tickets-were-sold-for-the-first-game-ge",{"id":95,"category":45,"text_question":96,"photo_question":47,"text_answer":97,"step_text_answer":8,"step_photo_answer":8,"views":98,"likes":99,"slug":100},537623,"There is a $1,800 difference between the prices of two items. If the prices are in the ratio 3:4,\n what are the prices of the items?","Dado que los precios de los artículos están en la razón 3:4, establecemos una ecuación para representar esta relación: \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{x}{y} = \\frac{3}{4}\u003C/math-field>\u003C/math-field>. Reescribimos esta ecuación como \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = \\frac{4}{3}x\u003C/math-field>\u003C/math-field> y la sustituimos en la expresión de la diferencia de los precios \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y - x = 1800\u003C/math-field>\u003C/math-field>. Esto produce \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{4}{3}x - x = 1800\u003C/math-field>\u003C/math-field>, que simplificamos a \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>\\frac{1}{3}x = 1800\u003C/math-field>\u003C/math-field>. Al despejar para \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x\u003C/math-field>\u003C/math-field>, encontramos \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>x = 5400\u003C/math-field>\u003C/math-field>. Luego sustituimos este valor en la ecuación para \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y\u003C/math-field>\u003C/math-field> para descubrir que \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>y = 7200\u003C/math-field>\u003C/math-field>. Así, los precios de los dos artículos son \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>5400\u003C/math-field>\u003C/math-field> y \u003Cmath-field read-only default-mode=\"inline-math\" class=\"math-expression\">\u003Cmath-field read-only>7200\u003C/math-field>\u003C/math-field>.",992,198,"there-is-a-1-800-difference-between-the-prices-of-two-items-if-the-prices-are-in-the-ratio-3-4-what-are-the-prices-of-the-items",{"id":102,"category":45,"text_question":103,"photo_question":47,"text_answer":104,"step_text_answer":8,"step_photo_answer":8,"views":105,"likes":106,"slug":107},537562,"1/3+3/10+2/15","Apply the fraction rule \u003Cmath-field read-only default-mode=\"inline-math\" style=\"padding-left:5px; padding-right:5px; display:inline-block; border-radius:4px; border: 1px solid rgba(0, 0, 0, .3)\">$\\frac{a}{c}+\\frac{b}{c}=\\frac{a+b}{c}$\u003C/math-field>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$=\\frac{10+9+4}{30}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>\n \n Add the numbers: \u003Cmath-field read-only default-mode=\"inline-math\" style=\"padding-left:5px; padding-right:5px; display:inline-block; border-radius:4px; border: 1px solid rgba(0, 0, 0, .3)\">$ 10+9+4=23$\u003C/math-field>\n \n \u003Cdiv>\n \n \u003Cmath-field style=\"font-size: 16px;padding: 8px;border-radius: 8px;border: 1px solid rgba(0, 0, 0, .3);box-shadow: 0 0 0 rgba(0, 0, 0, .2)\n\" read-only>$=\\frac{23}{30}$\u003C/math-field>\n \u003Cbr>\n \u003C/div>",1005,201,"1-3-3-10-2-15",{"id":109,"category":45,"text_question":110,"photo_question":47,"text_answer":111,"step_text_answer":8,"step_photo_answer":8,"views":112,"likes":113,"slug":114},537552,"The company's coverage is 10 times (Bat/interest), its accounting profit is 200 million and its income tax rate is 20%, so its interest is:\n\n\n a. None of the above\n\n\n b. 250 million.\n\n\n c. 2.5 million.\n\n\n d. 25 million.\n\n Only considers important and relevant data","Respuesta = Para encontrar el interés de la empresa en base a la información proporcionada, podemos utilizar la siguiente relación:\n\n 1. **Entendiendo el índice de cobertura**: El índice de cobertura indica cuántas veces las ganancias pueden cubrir los gastos de intereses. En este caso, la cobertura se expresa como 10 veces.\n\n 2. **Resultado Contable**: El resultado contable (antes de impuestos) es de 200 millones.\n\n 3. **Tasa de impuesto**: La tasa de impuesto es del 20%.\n\n ### Paso 1: Calcular el ingreso neto después de impuestos\n\n El ingreso neto después de impuestos se puede calcular utilizando la fórmula:\n\n $$\\[\\text{Ingreso neto} = \\text{Utilidad contable} \\times (1 - \\text{Tasa impositiva})\\]44\n\n Sustituyendo los valores:\n\n \u003Cmath-field read-only>\\[\\text{Ingreso neto} = 200 \\, \\text{millones} \\times (1 - 0,20) = 200 \\, \\text{millones} \\times 0,80 = 160 \\, \\text{millones}\\]\u003C/math-field>\n\n ### Paso 2: Relacione el ingreso neto con el interés\n\n Como la tasa de cobertura es 10 veces, podemos expresarla como:\n\n \u003Cmath-field read-only>\\[\\text{Ratio de cobertura} = \\frac{\\text{Ingreso neto}}{\\text{Interés}} = 10\\]\u003C/math-field>\n\n A partir de esto, podemos reorganizar para encontrar el interés:\n\n \u003Cmath-field read-only>\\[\\text{Interés} = \\frac{\\text{Ingreso neto}}{\\text{Ratio de cobertura}}\\]\u003C/math-field>\n\n Sustituyendo los valores que calculamos:\n\n \u003Cmath-field read-only>\\[\\text{Interés} = \\frac{160 \\, \\text{millones}}{10} = 16 \\, \\text{millones}\\]\u003C/math-field>\n\n ### Paso 3: Conclusión\n\n Dado que \u003Cmath-field read-only>\$16 \\, \\text{millones}\$\u003C/math-field> no es una de las opciones proporcionadas (250 millones, 2,5 millones, 25 millones o ninguna de las anteriores), la respuesta correcta sería:\n\n **a. Ninguna de las anteriores.**\n\n Si tienes más preguntas o necesitas alguna aclaración, ¡no dudes en 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