Solve the simultaneous equations:Multiply the second equation through by 5 to make the -terms cancel when we add the equations vertically term by term: Adding them term by term: Divide both sides by 67 Factor: Substitute into Divide both sides by : So we have these solutions for , (x,y) = (6,8) (x,y) = (6,-8) Substitute into Divide both sides by : So we have these solutions for , (x,y) = (-6,8) (x,y) = (-6,-8) So there are four solutions: (x,y) = (6,8) (x,y) = (6,-8) (x,y) = (-6,8) (x,y) = (-6,-8) ----------------------- What is the substitution for the quadratic formula for equation: Since this already contains a small letter "a", we'll write the quadratic formula using all CAPITAL LETTERS: where , , and Using the +, Using the +, ------------------------------ What would the problem like at the beginning: To clear of fractions, multiply second equation through by 2: Solve the second equation for y: Substitute that in the first equation: Substitute each of those into: So there are 4 solutions: (x,y)=(7,4) (x,y)=(-7,-4) (x,y)=(4,7) (x,y)=(-4,-7) ---------------------- if the discriminant of a complete quadratic equation is 8, what is the nature of its roots. When the discriminant is positive, there are two different real solutions. When it is 0 there is exactly one real solution When it is negative there are no real solutions. 8 is positive so there are two different real solutions. ----------------------- separate 72 into two parts so that the first part is the square of the second. let x represent the first part, what is the equation that would solve the problem. The other part of is First part: Second part: >>...the first part is the square of the second part...<< So So if x = first part = 81, then second part = 72-81=-9 That's kind of weird, separating 72 into 81 and -9, but they do sum to 72, and certainly 81 is the square of -9. Now if x = first part = 64, then second part = 72-64=8 That's not weird, separating 72 into 64 and 8, and they do sum to 72, and certainly 64 is the square of 8. So there are two answers, a weird one and one not weird at all. Edwin