Question

Select the correct difference.

5d² + 4d - 3 less 3d² -2d + 4

Answer 1

`Solution,solve\:for\:d,\:5d^2+4d<3d^2-2d+4\quad :\quad (-3-√(17))/(2)<d<(√(17)-3)/(2)`

`Steps:`

`\mathrm{Subtract\:}3d^2-2d+4\mathrm{\:from\:both\:sides}, 5d^2+4d-\left(3d^2-2d+4\right)<3d^2-2d+4-\left(3d^2-2d+4\right)`

`\mathrm{Refine}, 2d^2+6d-4<0`

`\mathrm{Divide\:both\:sides\:by\:}2, d^2+3d-2<0`

`d^2+3d-2=0\quad :\quad d=(-3+√(17))/(2),\:d=(-3-√(17))/(2)`

`\mathrm{Factor\:into\:the\:form}\:\left(x-a\right)\left(x-b\right), \left(d-(-3+√(17))/(2)\right)\left(d-(-3-√(17))/(2)\right)<0`

`\mathrm{Compute\:the\:signs\:of\:the\:factors\:of\:}\left(d-(-3+√(17))/(2)\right)\left(d-(-3-√(17))/(2)\right)`

`\mathrm{Compute\:the\:signs\:of\:}d-(-3+√(17))/(2), \mathrm{Compute\:the\:signs\:of\:}d-(-3-√(17))/(2)`

`\mathrm{Choosing\:ranges\:that\:satisfy\:the\:required\:condition:}\:<\:0, (-3-√(17))/(2)<d<(√(17)-3)/(2)`

The correct answer is
`(-3-√(17))/(2)<d<(√(17)-3)/(2)`