Question

The sides of a triangle are x,x+1,2x-1 and its area is x root 10. Find the value of x

Answer 1

You can use Heron's formula to solve this problem.

$A= \sqrt{s(s-a)(s-b)(s-c)}$ , where A is the area; a,b,с are the sides;  s is the semiperimeter of the triangle
$s= \frac{a+b+c}{2}$

According to conditions:
a = x
b = x+1
с = 2x-1

Calculate the semiperimeter:
$s= \frac{x+x+1+2x-1}{2} = \frac{4x}{2}=2x$

Now we can compose and solve the equation according to Heron's formula:
$\sqrt{2x(2x-x)(2x-(x+1))(2x-(2x-1))}=x \sqrt{10} \\ \sqrt{2x(x)(2x-x-1)(2x-2x+1)}=x \sqrt{10} \\ \sqrt{2x^2(x-1)(1)}=x \sqrt{10} \\ \sqrt{2x^3-2x^2}=x \sqrt{10} \ \ \ |()^2 \\ 2x^3-2x^2=10x^2 \\ 2x^3-2x^2-10x^2=0 \\ 2x^3-12x^2=0 \ \ |:2 \\ x^3-6x^2=0 \\ x^2(x-6)=0 \\ \\ x^2=0 \\ x=0 \ \ \ \O \\ or \\ x-6=0 \\ x=6$

As a result, x = 6 is your answer.
 
I hope this helped.