Question

What two numbers add up to 13 and multiply to -24

Answer 1

`\large\boxed{x=(13-√(265))/(2)\ and\ y=(13+√(265))/(2)}\\\\.\qquad\qquad\qquad\qquad or\\\\\boxed{x=(13+√(265))/(2)\ and\ y=(13-√(265))/(2)}`

Explanation:

`x,\ y-two\ numbers\\\\\text{We have the equations}\\\\x+y=13\ \text{and}\ xy=-24\\\\x+y=13\qquad\text{subtract y from both sides}\\x=13-y\qquad\text{substitute it to the second equation}\\\\(13-y)y=-24\qquad\text{use distributive property}\\13y-y^2=-24\qquad\text{add 24 to both sides}\\-y^2+13y+24=0\qquad\text{change the signs}\\y^2-13y-24=0`

Use the quadratic formula:

`ax^2+bx+c=0\\\\\Delta=b^2-4ac\\\\x_1=(-b-\sqrt\Delta)/(2a),\ x_2=(-b+\sqrt\Delta)/(2a)`

`y^2-13y-24=0\\\\a=1,\ b=-13,\ c=-24\\\\\Delta=(-13)^2-4(1)(-24)=169+96=265\\\\y_1=(-(-13)-√(265))/(2(1))=(13-√(265))/(2)\\\\y_2=(-(-13)+√(265))/(2(1))=(13+√(265))/(2)`

Put the values of y to the equation x = 13 - y:

`x_1=13-(13-√(265))/(2)=(26)/(2)-(13-√(256))/(2)=(26-13-(-√(256)))/(2)=(13+√(265))/(2)\\\\x_2=13-(13+√(265))/(2)=(26)/(2)-(13+√(265))/(2)=(26-13-√(256))/(2)=(13-√(265))/(2)`