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Created on: 2012-02-16

A sample problem solved by Quickmath web math solver

study how to multiply out a sum of 3 terms. ((a+b)/2)^2-(a-b)^2

Command

Solve

Equation
x^4-2*x^3-x = 0

  1. an equation Left side of the equation is equal to a sum consisting of 3 terms; the 1st term of the sum is equal to a power; the base is x; the exponent is four; the 2nd term of the sum is a negative product of 2 factors; the 1st factor of the product is equal to two; the 2nd factor of the product is equal to a power; the base is x; the exponent is three; the 3rd term of the sum is equal to negative x Right side of the equation is equal to zero;
  2. x to the power of four plus negative two multiplied by x raised to the power of three plus negative x is equal to zero.
Variable
Result

Exact

Solution 1 ( complex )
x = 4*(sqrt(3)*%i/2-1/2)/(9*(sqrt(59)/(2*3^(3/2))+43/54)^(1/3))+(sqrt(59)/(2*3^(3/2))+43/54)^(1/3)*(-sqrt(3)*%i/2-1/2)+2/3

  1. an equation Left side of the equation is x RHS is a sum comprising 3 terms. The 1st term of the sum is equal to a rational expression: the numerator of the rational expression is a product that consists of 2 factors. The 1st factor of the product is equal to four. The 2nd factor of the product is equal to a sum that contains 2 terms. The 1st term of the sum is equal to a fraction: the numerator of the fraction is a product comprising 2 factors. The 1st factor of the product is equal to square root of three. The 2nd factor of the product is i. The denominator of the fraction is two. The 2nd term of the sum is equal to a negative fraction: the top of the fraction is one. The bottom of the fraction is two. The denominator of the rational expression is a product consisting of 2 factors. The 1st factor of the product is nine. The 2nd factor of the product is equal to a power. The base is a sum that consists of 2 terms. The 1st term of the sum is a quotient: dividend of the quotient is square root of fifty nine. Divisor of the quotient is a product containing 2 factors. The 1st factor of the product is equal to two. The 2nd factor of the product is a power. The base is three. The exponent is a rational expression: the numerator of the rational expression is three. The denominator of the rational expression is two. The 2nd term of the sum is a quotient: dividend of the quotient is forty three. Divisor of the quotient is fifty four. The exponent is a quotient: dividend of the quotient is one. Divisor of the quotient is three. The 2nd term of the sum is equal to a product consisting of 2 factors. The 1st factor of the product is equal to a power. The base is a sum containing 2 terms. The 1st term of the sum is a fraction: the numerator of the fraction is square root of fifty nine. The denominator of the fraction is a product that comprises 2 factors. The 1st factor of the product is equal to two. The 2nd factor of the product is equal to a power. The base is three. The exponent is a quotient: dividend of the quotient is three. Divisor of the quotient is two. The 2nd term of the sum is equal to a fraction: the numerator of the fraction is forty three. The denominator of the fraction is fifty four. The exponent is a rational expression: the top of the rational expression is one. The bottom of the rational expression is three. The 2nd factor of the product is equal to a sum of 2 terms. The 1st term of the sum is equal to a fraction: the numerator of the fraction is a product containing 2 factors. The 1st factor of the product is equal to negative square root of three. The 2nd factor of the product is equal to i. The denominator of the fraction is two. The 2nd term of the sum is equal to a negative rational expression: the top of the rational expression is one. The bottom of the rational expression is two. The 3rd term of the sum is equal to a rational expression: the top of the rational expression is two. The bottom of the rational expression is three.
  2. x is four multiplied by open bracket square root of three multiplied by i divided by two plus negative one divided by two close bracket fraction line nine multiplied by left parenthesis square root of fifty nine divided by two multiplied by three to the power of three fraction line two plus forty three fraction line fifty four right parenthesis exponentiated by one fraction line three plus opening brace square root of fifty nine fraction line two multiplied by three to the power three divided by two plus forty three fraction line fifty four closing brace raised to the power one divided by three multiplied by left parenthesis negative square root of three times i fraction line two plus negative one over two right parenthesis plus two divided by three;
Solution 2 ( complex )
x = (sqrt(59)/(2*3^(3/2))+43/54)^(1/3)*(sqrt(3)*%i/2-1/2)+4*(-sqrt(3)*%i/2-1/2)/(9*(sqrt(59)/(2*3^(3/2))+43/54)^(1/3))+2/3

  1. an equation Left side of the equation is x RHS is a sum consisting of 3 terms; the 1st term of the sum is a product consisting of 2 factors; the 1st factor of the product is a power; the base is a sum consisting of 2 terms; the 1st term of the sum is equal to a fraction: the top of the fraction is square root of fifty nine; the bottom of the fraction is a product containing 2 factors; the 1st factor of the product is equal to two; the 2nd factor of the product is equal to a power; the base is three; the exponent is a rational expression: the top of the rational expression is three; the bottom of the rational expression is two; the 2nd term of the sum is a rational expression: the top of the rational expression is forty three; the bottom of the rational expression is fifty four; the exponent is a rational expression: the numerator of the rational expression is one; the denominator of the rational expression is three; the 2nd factor of the product is equal to a sum containing 2 terms; the 1st term of the sum is a rational expression: the numerator of the rational expression is a product that consists of 2 factors; the 1st factor of the product is equal to square root of three; the 2nd factor of the product is equal to i; the denominator of the rational expression is two; the 2nd term of the sum is equal to a negative rational expression: the top of the rational expression is one; the bottom of the rational expression is two; the 2nd term of the sum is a rational expression: the top of the rational expression is a product comprising 2 factors; the 1st factor of the product is equal to four; the 2nd factor of the product is a sum consisting of 2 terms; the 1st term of the sum is a fraction: the numerator of the fraction is a product consisting of 2 factors; the 1st factor of the product is equal to negative square root of three; the 2nd factor of the product is equal to i; the denominator of the fraction is two; the 2nd term of the sum is a negative rational expression: the top of the rational expression is one; the bottom of the rational expression is two; the bottom of the rational expression is a product comprising 2 factors; the 1st factor of the product is equal to nine; the 2nd factor of the product is a power; the base is a sum comprising 2 terms; the 1st term of the sum is a fraction: the top of the fraction is square root of fifty nine; the bottom of the fraction is a product consisting of 2 factors; the 1st factor of the product is equal to two; the 2nd factor of the product is equal to a power; the base is three; the exponent is a quotient: dividend of the quotient is three; divisor of the quotient is two; the 2nd term of the sum is a rational expression: the numerator of the rational expression is forty three; the denominator of the rational expression is fifty four; the exponent is a quotient: dividend of the quotient is one; divisor of the quotient is three; the 3rd term of the sum is equal to a rational expression: the numerator of the rational expression is two; the denominator of the rational expression is three;
  2. x is equal to opening parenthesis square root of fifty nine divided by two multiplied by three raised to the power of three divided by two plus forty three fraction line fifty four closing parenthesis to the power of one fraction line three multiplied by opening bracket square root of three times i divided by two plus negative one fraction line two closing bracket plus four multiplied by open bracket negative square root of three multiplied by i over two plus negative one divided by two close bracket fraction line nine times open bracket square root of fifty nine over two times three raised to the power three divided by two plus forty three fraction line fifty four close bracket raised to the power one over three plus two fraction line three.
Solution 3 ( real )
x = (sqrt(59)/(2*3^(3/2))+43/54)^(1/3)+4/(9*(sqrt(59)/(2*3^(3/2))+43/54)^(1/3))+2/3

  1. an equation LHS is x Right side of the equation is a sum that consists of 3 terms; the 1st term of the sum is a power; the base is a sum that consists of 2 terms; the 1st term of the sum is equal to a rational expression: the numerator of the rational expression is square root of fifty nine; the denominator of the rational expression is a product consisting of 2 factors; the 1st factor of the product is two; the 2nd factor of the product is equal to a power; the base is three; the exponent is a rational expression: the numerator of the rational expression is three; the denominator of the rational expression is two; the 2nd term of the sum is equal to a quotient: dividend of the quotient is forty three; divisor of the quotient is fifty four; the exponent is a rational expression: the top of the rational expression is one; the bottom of the rational expression is three; the 2nd term of the sum is equal to a rational expression: the numerator of the rational expression is four; the denominator of the rational expression is a product that comprises 2 factors; the 1st factor of the product is nine; the 2nd factor of the product is a power; the base is a sum that comprises 2 terms; the 1st term of the sum is equal to a fraction: the top of the fraction is square root of fifty nine; the bottom of the fraction is a product of 2 factors; the 1st factor of the product is equal to two; the 2nd factor of the product is equal to a power; the base is three; the exponent is a fraction: the top of the fraction is three; the bottom of the fraction is two; the 2nd term of the sum is equal to a rational expression: the numerator of the rational expression is forty three; the denominator of the rational expression is fifty four; the exponent is a fraction: the top of the fraction is one; the bottom of the fraction is three; the 3rd term of the sum is equal to a rational expression: the numerator of the rational expression is two; the denominator of the rational expression is three;
  2. x is equal to opening brace square root of fifty nine fraction line two times three to the power of three over two plus forty three fraction line fifty four closing brace raised to the power one fraction line three plus four over nine times open parenthesis square root of fifty nine over two times three raised to the power of three over two plus forty three fraction line fifty four close parenthesis raised to the power one over three plus two divided by three.
Solution 4 ( real )
x = 0

  1. an equation Left side of the equation is x RHS is zero.
  2. x is zero.


Approximate

Solution 1 ( complex )
x = -.665457*%i-.102785

  1. an equation LHS is x Right side of the equation is equal to a sum comprising 2 terms. The 1st term of the sum is a product that comprises 2 factors. The 1st factor of the product is equal to negative zero point six tenth, six hundredth, five ten hundredth, four thousandth, five ten thousandth and seven hundred thousandth. The 2nd factor of the product is i. The 2nd term of the sum is equal to negative zero point one zero two seven eight five.
  2. x is equal to negative zero point six tenth, six hundredth, five ten hundredth, four thousandth, five ten thousandth and seven hundred thousandth times i plus negative zero point one tenth, two ten hundredth, seven thousandth, eight ten thousandth and five hundred thousandth;
Solution 2 ( complex )
x = .665457*%i-.102785

  1. an equation Left side of the equation is x Right side of the equation is a sum that contains 2 terms; the 1st term of the sum is equal to a product that consists of 2 factors; the 1st factor of the product is equal to zero point six six five four five seven; the 2nd factor of the product is i; the 2nd term of the sum is negative zero point one tenth, two ten hundredth, seven thousandth, eight ten thousandth and five hundred thousandth;
  2. x is zero point six tenth, six hundredth, five ten hundredth, four thousandth, five ten thousandth and seven hundred thousandth multiplied by i plus negative zero point one tenth, two ten hundredth, seven thousandth, eight ten thousandth and five hundred thousandth.
Solution 3 ( real )
x = 2.20557

  1. an equation Left side of the equation is equal to x RHS is two point two tenth, five ten hundredth, five thousandth and seven ten thousandth.
  2. x is equal to two point two tenth, five ten hundredth, five thousandth and seven ten thousandth.
Solution 4 ( real )
x = 0.0

  1. an equation LHS is x Right side of the equation is zero.
  2. x is equal to zero;