Question 645496
1. The height of a triangle {{{h}}} is {{{2m}}} more than twice the length of the base {{{a}}}.

=>... {{{h=2a+2m}}}...=>..{{{h-2m=2a}}} =>..{{{a=(h-2m)/2}}}

The area of the triangle {{{A}}} is {{{A=42m^2}}}. 

Find the height of the triangle.

{{{A=(1/2)a*h}}}

{{{42m^2=(1/2)a*h}}}........replace {{{a}}} with {{{(h-2m)/2}}}

{{{42m^2=(1/2)(h-2m)/2*h}}}...solve for {{{h}}}

{{{42m^2=((h-2m)/4)*h}}}

{{{42m^2=((h-2m)h/4)}}}

{{{42m^2=((h^2-2hm)/4)}}}

{{{42m^2*4=h^2-2hm}}}

{{{168m^2=h^2-2hm}}}

{{{0=h^2-2hm-168m^2}}}...or

{{{h^2-2hm-168m^2=0}}}....use quadratic formula to find {{{h}}}

{{{h = (-(-2) +- sqrt( (-2)^2-4*1*(-168) ))/(2*1) }}}

{{{h = (2 +- sqrt(4+672 ))/2 }}}

{{{h = (2 +- sqrt(676 ))/2 }}}

{{{h = (2 +- 26)/2 }}}

solution: we need only positive solution since the height cannot be negative

{{{h = (2 + 26)/2 }}}

{{{h = 28/2 }}}

{{{highlight(h = 14m )}}}

now find the length of the base {{{a}}}

{{{a=(h-2m)/2}}}

{{{a=(14m-2m)/2}}}

{{{a=(12m)/2}}}

{{{highlight(a=6m)}}}


2.

{{{a^2-3a = 28}}} 

{{{a^2-3a -28=0}}} 

{{{a = (-(-3) +- sqrt( (-3)^2-4*1*(-28) ))/(2*1) }}}

{{{a = (3 +- sqrt(9+112))/2 }}}

{{{a = (3 +- sqrt(121))/2 }}}

{{{a = (3 +- 11)/2 }}}

solutions:

{{{a = (3 +11)/2 }}}

{{{a = 17/2 }}}

{{{highlight(a = 7) }}}

or

{{{a = (3 -11)/2 }}}

{{{a = -8/2 }}}

{{{highlight(a = -4 )}}}




3.

{{{(a -4)(a + 6) =-16}}}
 
{{{a^2+6a -4a -24 =-16}}}

{{{a^2+2a-24 +16=0}}}

{{{a^2+2a-8=0}}}

{{{a = (-2 +- sqrt( 2^2-4*1*(-8) ))/(2*1) }}}

{{{a = (-2 +- sqrt( 4+32 ))/2 }}}

{{{a = (-2 +- sqrt( 36 ))/2 }}}

{{{a = (-2 +- 6)/2 }}}

solutions:

{{{a = (-2 + 6)/2 }}}

{{{a = 4/2 }}}

{{{highlight(a =2) }}}

or


{{{a = (-2- 6)/2 }}}

{{{a = -8/2 }}}

{{{highlight(a =-4)}}}
 
 
4. 


{{{5y(y + 6) = 4(y + 6)}}}...first multiply

 {{{5y^2 + 30 = 4y + 24}}}

{{{5y^2 + 30 -4y -24=0}}}

{{{5y^2 -4y+ 6 =0}}}

 {{{y = (-(-4) +- sqrt( (-4)^2-4*5*6 ))/(2*5) }}}

{{{y = (4 +- sqrt(16-120 ))/10 }}}

{{{y = (4 +- sqrt(-104 ))/10 }}}

{{{y = (4 +- sqrt((-1)104 ))/10 }}}

{{{y = (4 +- i*sqrt(104 ))/10 }}}

{{{y = (4 +- 10.2i)/10 }}}

solutions:

{{{y = (4 + 10.2i)/10 }}}

{{{y = 4/10 + 10.2i/10 }}}

{{{highlight(y = 0.4+ 1.02i) }}}

or

{{{y = (4 - 10.2i)/10 }}}

{{{y = 4/10 - 10.2i/10 }}}

{{{highlight(y = 0.4- 1.02i )}}}