Factor out the common term \(x\).
\[\frac{x(x+3)}{2}=\frac{x+7}{4}\]
Multiply both sides by \(4\) (the LCM of \(2, 4\)).
\[2x(x+3)=x+7\]
Expand.
\[2{x}^{2}+6x=x+7\]
Move all terms to one side.
\[2{x}^{2}+6x-x-7=0\]
Simplify \(2{x}^{2}+6x-x-7\) to \(2{x}^{2}+5x-7\).
\[2{x}^{2}+5x-7=0\]
Split the second term in \(2{x}^{2}+5x-7\) into two terms.
Multiply the coefficient of the first term by the constant term.
\[2\times -7=-14\]
Ask: Which two numbers add up to \(5\) and multiply to \(-14\)?
Split \(5x\) as the sum of \(7x\) and \(-2x\).
\[2{x}^{2}+7x-2x-7\]
\[2{x}^{2}+7x-2x-7=0\]
Factor out common terms in the first two terms, then in the last two terms.
\[x(2x+7)-(2x+7)=0\]
Factor out the common term \(2x+7\).
\[(2x+7)(x-1)=0\]
Solve for \(x\).
Ask: When will \((2x+7)(x-1)\) equal zero?
When \(2x+7=0\) or \(x-1=0\)
Solve each of the 2 equations above.
\[x=-\frac{7}{2},1\]
\[x=-\frac{7}{2},1\]
Decimal Form: -3.5, 1
x=-7/2,1
