Solve (7+x)(4+x)/2=k

$$\frac { ( 7 + x ) ( 4 + x ) } { 2 } = k$$

Solve for k

$$k=\frac{\left(x+4\right)\left(x+7\right)}{2}$$

Solution Steps

Use the distributive property to multiply 7+x by 4+x and combine like terms.

$$\frac{28+11x+x^{2}}{2}=k$$

Divide each term of 28+11x+x^{2} by 2 to get 14+\frac{11}{2}x+\frac{1}{2}x^{2}.

$$14+\frac{11}{2}x+\frac{1}{2}x^{2}=k$$

Swap sides so that all variable terms are on the left hand side.

$$k=14+\frac{11}{2}x+\frac{1}{2}x^{2}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{8k+9}-11}{2}$$

$$x=\frac{-\sqrt{8k+9}-11}{2}$$

Solve for x

$$x=\frac{\sqrt{8k+9}-11}{2}$$

$$x=\frac{-\sqrt{8k+9}-11}{2}\text{, }k\geq -\frac{9}{8}$$