Answer:
-1 or 5/2
Explanation:
The ratio in the question implies that, by cross-multiplying, x2⋅2=(3x+5)⋅1 , in other words 2x2=3x+5 . Therefore, 2x2−3x−5=0 .
There are many ways to try to solve this, one way is to factor: 2x2−3x−5=(x+1)(2x−5) , so the possible values of x are −1 and 52 .
Another way is to use the quadratic formula: x=−b±b2−4ac√2a=3±32+4⋅2⋅5√2⋅2=3±49√4=3±74 , which is either −1 or 52 .
Another way is completing the square:
2x2−3x−5=0
Dividing both sides by 2, x2−32x−52=0
Adding 52 to separate summands, x2−32x=52
Adding 916 to complete the square, x2−32x+916=52+916=4916
Factoring the left-hand side gives (x−34)2=4916 . Hence, taking square roots, x−34=±74 .
Moreover, x=34±74 , which is either −1 or 52 .
Another way is inverting the square:
2x2−3x−5=0⟹2x−5x=3
(2x−5x)2=9
4x2−20+25x2=9 (expand left-hand side)
4x2+20+25x2=49 (add 40 to both sides)
(2x+5x)2=49 (factor left-hand side)
2x+5x=±7
4x=(2x−5x)+(2x+5x)=3±7
x=3±74 , which is either −1 or 52 .
Another way is to graph; that strategy will be omitted here.