Question 345414
x^(2)+(x+70)^(2)=170^(2)

Squaring a number is the same as multiplying the number by itself (170*170).  In this case, 170 squared is 28900.
x^(2)+(x+70)^(2)=28900

To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
x^(2)+(x+70)^(2)-28900=0

Squaring an expression is the same as multiplying the expression by itself 2 times.
x^(2)+(x+70)(x+70)-28900=0

Multiply each term in the first group by each term in the second group using the FOIL method. FOIL stands for First Outer Inner Last, and is a method of multiplying two binomials.  First, multiply the first two terms in each binomial group.  Next, multiply the outer terms in each group, followed by the inner terms.  Finally, multiply the last two terms in each group.
x^(2)+(x*x+x*70+70*x+70*70)-28900=0

Simplify the FOIL expression by multiplying and combining all like terms.
x^(2)+(x^(2)+140x+4900)-28900=0

Since x^(2) and x^(2) are like terms, add x^(2) to x^(2) to get 2x^(2).
2x^(2)+140x+4900-28900=0

Subtract 28900 from 4900 to get -24000.
2x^(2)+140x-24000=0

Factor out the GCF of 2 from each term in the polynomial.
2(x^(2))+2(70x)+2(-12000)=0

Factor out the GCF of 2 from 2x^(2)+140x-24000.
2(x^(2)+70x-12000)=0

In this problem 150*-80=-12000 and 150-80=70, so insert 150 as the right hand term of one factor and -80 as the right-hand term of the other factor.
2(x+150)(x-80)=0

Divide both sides of the equation by 2.  Dividing 0 by any non-zero number is 0.
(x+150)(x-80)=0

Set each of the factors of the left-hand side of the equation equal to 0.
x+150=0_x-80=0

Since 150 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 150 from both sides.
x=-150_x-80=0

Set each of the factors of the left-hand side of the equation equal to 0.
x=-150_x-80=0

Since -80 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 80 to both sides.
x=-150_x=80

The complete solution is the set of the individual solutions.
x=-150,80

Answer= One has traveled 80 miles and the other 70 miles; they are 150 miles apart