Solve 2sqrt{(2x+8)(x+7)}=36-3x

$$2 \sqrt { ( 2 x + 8 ) ( x + 7 ) } = 36 - 3 x$$

Solve for x

$$x=152-36\sqrt{17}\approx 3.568197478$$

Solution Steps

Square both sides of the equation.

$$\left(2\sqrt{\left(2x+8\right)\left(x+7\right)}\right)^{2}=\left(36-3x\right)^{2}$$

Use the distributive property to multiply 2x+8 by x+7 and combine like terms.

$$\left(2\sqrt{2x^{2}+22x+56}\right)^{2}=\left(36-3x\right)^{2}$$

Expand \left(2\sqrt{2x^{2}+22x+56}\right)^{2}.

$$2^{2}\left(\sqrt{2x^{2}+22x+56}\right)^{2}=\left(36-3x\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{2x^{2}+22x+56}\right)^{2}=\left(36-3x\right)^{2}$$

Calculate \sqrt{2x^{2}+22x+56} to the power of 2 and get 2x^{2}+22x+56.

$$4\left(2x^{2}+22x+56\right)=\left(36-3x\right)^{2}$$

Use the distributive property to multiply 4 by 2x^{2}+22x+56.

$$8x^{2}+88x+224=\left(36-3x\right)^{2}$$

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

$$8x^{2}+88x+224=1296-216x+9x^{2}$$

Subtract 1296 from both sides.

$$8x^{2}+88x+224-1296=-216x+9x^{2}$$

Subtract 1296 from 224 to get -1072.

$$8x^{2}+88x-1072=-216x+9x^{2}$$

Add 216x to both sides.

$$8x^{2}+88x-1072+216x=9x^{2}$$

Combine 88x and 216x to get 304x.

$$8x^{2}+304x-1072=9x^{2}$$

Subtract 9x^{2} from both sides.

$$8x^{2}+304x-1072-9x^{2}=0$$

Combine 8x^{2} and -9x^{2} to get -x^{2}.

$$-x^{2}+304x-1072=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 304 for b, and -1072 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-304±\sqrt{304^{2}-4\left(-1\right)\left(-1072\right)}}{2\left(-1\right)}$$

Square 304.

$$x=\frac{-304±\sqrt{92416-4\left(-1\right)\left(-1072\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-304±\sqrt{92416+4\left(-1072\right)}}{2\left(-1\right)}$$

Multiply 4 times -1072.

$$x=\frac{-304±\sqrt{92416-4288}}{2\left(-1\right)}$$

Add 92416 to -4288.

$$x=\frac{-304±\sqrt{88128}}{2\left(-1\right)}$$

Take the square root of 88128.

$$x=\frac{-304±72\sqrt{17}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-304±72\sqrt{17}}{-2}$$

Now solve the equation x=\frac{-304±72\sqrt{17}}{-2} when ± is plus. Add -304 to 72\sqrt{17}.

$$x=\frac{72\sqrt{17}-304}{-2}$$

Divide -304+72\sqrt{17} by -2.

$$x=152-36\sqrt{17}$$

Now solve the equation x=\frac{-304±72\sqrt{17}}{-2} when ± is minus. Subtract 72\sqrt{17} from -304.

$$x=\frac{-72\sqrt{17}-304}{-2}$$

Divide -304-72\sqrt{17} by -2.

$$x=36\sqrt{17}+152$$

The equation is now solved.

$$x=152-36\sqrt{17} x=36\sqrt{17}+152$$

Substitute 152-36\sqrt{17} for x in the equation 2\sqrt{\left(2x+8\right)\left(x+7\right)}=36-3x.

$$2\sqrt{\left(2\left(152-36\sqrt{17}\right)+8\right)\left(152-36\sqrt{17}+7\right)}=36-3\left(152-36\sqrt{17}\right)$$

Simplify. The value x=152-36\sqrt{17} satisfies the equation.

$$108\times 17^{\frac{1}{2}}-420=-420+108\times 17^{\frac{1}{2}}$$

Substitute 36\sqrt{17}+152 for x in the equation 2\sqrt{\left(2x+8\right)\left(x+7\right)}=36-3x.

$$2\sqrt{\left(2\left(36\sqrt{17}+152\right)+8\right)\left(36\sqrt{17}+152+7\right)}=36-3\left(36\sqrt{17}+152\right)$$

Simplify. The value x=36\sqrt{17}+152 does not satisfy the equation because the left and the right hand side have opposite signs.

$$108\times 17^{\frac{1}{2}}+420=-420-108\times 17^{\frac{1}{2}}$$

Equation 2\sqrt{\left(x+7\right)\left(2x+8\right)}=36-3x has a unique solution.

$$x=152-36\sqrt{17}$$