Answer :
Answer:
4
Step-by-step explanation:
Angle CAB is opposite to the side BC.
We are going to use law of cosines to see this up since we are either given a numerical value or an algebraic expression per each side, and that we are also given an angle measurement for one of the 3 angles of our triangle.
[tex](2\sqrt{7})^2=x^2+(x+2)^2-2x(x+2)\cos(60^\circ)[/tex]
Simplify and use foil to multiply/expand [tex](x+2)^2[/tex]:
[tex]4(7)=x^2+(x+2)(x+2)-2x(x+2)\cos(60^\circ)[/tex]
[tex]28=x^2+x^2+4x+4-2x(x+2)\cos(60^\circ)[/tex]
Using [tex]\cos(60^\circ)=\frac{1}{2}[/tex] and doing a little simplifying gives:
[tex]28=2x^2+4x+4-2x(x+2)\frac{1}{2}[/tex]
Since 2(1/2)=1 we have:
[tex]28=2x^2+4x+4-x(x+2)[/tex]
Distribute:
[tex]28=2x^2+4x+4-x^2-2x[/tex]
Subtract 28 on both sides and simplify:
[tex]0=x^2+2x-24[/tex]
Find two numbers that multiply to be -24 and add to be 2. Those numbers are 6 and -4 so the factored form of the right hand is given in the equation:
[tex]0=(x+6)(x-4)[/tex]
This implies x+6=0 or x-4=0.
Solving the first equation for x gives us x=-6 (subtracted 6 on both sides).
Solving the second equation for x gives us x=4 (added 4 on both sides).
4 is the only one that can be a solution since a side can't have a negative measurement.
The answer is x=4.
