Given that x^2 : (3x+5) =1:2 find the possible values of x.

-1 or 5/2

Step-by-step 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.

middle maybe?

step-by-step explanation: