Solve sqrt{(2x+8)(x+5)}=36-3x

$$\sqrt { ( 2 x + 8 ) ( x + 5 ) } = 36 - 3 x$$

Solve for x

$$x = \frac{117 - \sqrt{4897}}{7} \approx 6.717347407$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{\left(2x+8\right)\left(x+5\right)}\right)^{2}=\left(36-3x\right)^{2}$$

Use the distributive property to multiply 2x+8 by x+5 and combine like terms.

$$\left(\sqrt{2x^{2}+18x+40}\right)^{2}=\left(36-3x\right)^{2}$$

Calculate \sqrt{2x^{2}+18x+40} to the power of 2 and get 2x^{2}+18x+40.

$$2x^{2}+18x+40=\left(36-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(36-3x\right)^{2}.

$$2x^{2}+18x+40=1296-216x+9x^{2}$$

Subtract 1296 from both sides.

$$2x^{2}+18x+40-1296=-216x+9x^{2}$$

Subtract 1296 from 40 to get -1256.

$$2x^{2}+18x-1256=-216x+9x^{2}$$

Add 216x to both sides.

$$2x^{2}+18x-1256+216x=9x^{2}$$

Combine 18x and 216x to get 234x.

$$2x^{2}+234x-1256=9x^{2}$$

Subtract 9x^{2} from both sides.

$$2x^{2}+234x-1256-9x^{2}=0$$

Combine 2x^{2} and -9x^{2} to get -7x^{2}.

$$-7x^{2}+234x-1256=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 234 for b, and -1256 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-234±\sqrt{234^{2}-4\left(-7\right)\left(-1256\right)}}{2\left(-7\right)}$$

Square 234.

$$x=\frac{-234±\sqrt{54756-4\left(-7\right)\left(-1256\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-234±\sqrt{54756+28\left(-1256\right)}}{2\left(-7\right)}$$

Multiply 28 times -1256.

$$x=\frac{-234±\sqrt{54756-35168}}{2\left(-7\right)}$$

Add 54756 to -35168.

$$x=\frac{-234±\sqrt{19588}}{2\left(-7\right)}$$

Take the square root of 19588.

$$x=\frac{-234±2\sqrt{4897}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-234±2\sqrt{4897}}{-14}$$

Now solve the equation x=\frac{-234±2\sqrt{4897}}{-14} when ± is plus. Add -234 to 2\sqrt{4897}.

$$x=\frac{2\sqrt{4897}-234}{-14}$$

Divide -234+2\sqrt{4897} by -14.

$$x=\frac{117-\sqrt{4897}}{7}$$

Now solve the equation x=\frac{-234±2\sqrt{4897}}{-14} when ± is minus. Subtract 2\sqrt{4897} from -234.

$$x=\frac{-2\sqrt{4897}-234}{-14}$$

Divide -234-2\sqrt{4897} by -14.

$$x=\frac{\sqrt{4897}+117}{7}$$

The equation is now solved.

$$x=\frac{117-\sqrt{4897}}{7} x=\frac{\sqrt{4897}+117}{7}$$

Substitute \frac{117-\sqrt{4897}}{7} for x in the equation \sqrt{\left(2x+8\right)\left(x+5\right)}=36-3x.

$$\sqrt{\left(2\times \frac{117-\sqrt{4897}}{7}+8\right)\left(\frac{117-\sqrt{4897}}{7}+5\right)}=36-3\times \frac{117-\sqrt{4897}}{7}$$

Simplify. The value x=\frac{117-\sqrt{4897}}{7} satisfies the equation.

$$-\left(\frac{99}{7}-\frac{3}{7}\times 4897^{\frac{1}{2}}\right)=-\frac{99}{7}+\frac{3}{7}\times 4897^{\frac{1}{2}}$$

Substitute \frac{\sqrt{4897}+117}{7} for x in the equation \sqrt{\left(2x+8\right)\left(x+5\right)}=36-3x.

$$\sqrt{\left(2\times \frac{\sqrt{4897}+117}{7}+8\right)\left(\frac{\sqrt{4897}+117}{7}+5\right)}=36-3\times \frac{\sqrt{4897}+117}{7}$$

Simplify. The value x=\frac{\sqrt{4897}+117}{7} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{99}{7}+\frac{3}{7}\times 4897^{\frac{1}{2}}=-\frac{99}{7}-\frac{3}{7}\times 4897^{\frac{1}{2}}$$

Equation \sqrt{\left(x+5\right)\left(2x+8\right)}=36-3x has a unique solution.

$$x=\frac{117-\sqrt{4897}}{7}$$