Solve sqrt{x+3-4sqrt{x-7}}+sqrt{x+8-6sqrt{x-7}}=7

$$\sqrt { x + 3 - 4 \sqrt { x - 7 } } + \sqrt { x + 8 - 6 \sqrt { x - 7 } } = 7$$

Solve for x

$$x=\frac{35\sqrt{2}}{4}+\frac{155}{8}\approx 31.749368671$$

$$x=-\frac{35\sqrt{2}}{4}+\frac{155}{8}\approx 7.000631329$$

Solution Steps

Subtract \sqrt{x+8-6\sqrt{x-7}} from both sides of the equation.

$$\sqrt{x+3-4\sqrt{x-7}}=7-\sqrt{x+8-6\sqrt{x-7}}$$

Square both sides of the equation.

$$\left(\sqrt{x+3-4\sqrt{x-7}}\right)^{2}=\left(7-\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Calculate \sqrt{x+3-4\sqrt{x-7}} to the power of 2 and get x+3-4\sqrt{x-7}.

$$x+3-4\sqrt{x-7}=\left(7-\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-\sqrt{x+8-6\sqrt{x-7}}\right)^{2}.

$$x+3-4\sqrt{x-7}=49-14\sqrt{x+8-6\sqrt{x-7}}+\left(\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Calculate \sqrt{x+8-6\sqrt{x-7}} to the power of 2 and get x+8-6\sqrt{x-7}.

$$x+3-4\sqrt{x-7}=49-14\sqrt{x+8-6\sqrt{x-7}}+x+8-6\sqrt{x-7}$$

Subtract 49+x+8-6\sqrt{x-7} from both sides of the equation.

$$x+3-4\sqrt{x-7}-\left(49+x+8-6\sqrt{x-7}\right)=-14\sqrt{x+8-6\sqrt{x-7}}$$

To find the opposite of 49+x+8-6\sqrt{x-7}, find the opposite of each term.

$$x+3-4\sqrt{x-7}-49-\left(x+8-6\sqrt{x-7}\right)=-14\sqrt{x+8-6\sqrt{x-7}}$$

Square both sides of the equation.

$$\left(x+3-4\sqrt{x-7}-49-\left(x+8-6\sqrt{x-7}\right)\right)^{2}=\left(-14\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Square x+3-4\sqrt{x-7}-49-\left(x+8-6\sqrt{x-7}\right).

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)+98\left(x+8-6\sqrt{x-7}\right)+2401=\left(-14\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Expand \left(-14\sqrt{x+8-6\sqrt{x-7}}\right)^{2}.

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)+98\left(x+8-6\sqrt{x-7}\right)+2401=\left(-14\right)^{2}\left(\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Calculate -14 to the power of 2 and get 196.

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)+98\left(x+8-6\sqrt{x-7}\right)+2401=196\left(\sqrt{x+8-6\sqrt{x-7}}\right)^{2}$$

Calculate \sqrt{x+8-6\sqrt{x-7}} to the power of 2 and get x+8-6\sqrt{x-7}.

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)+98\left(x+8-6\sqrt{x-7}\right)+2401=196\left(x+8-6\sqrt{x-7}\right)$$

Subtract 196\left(x+8-6\sqrt{x-7}\right) from both sides.

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)+98\left(x+8-6\sqrt{x-7}\right)+2401-196\left(x+8-6\sqrt{x-7}\right)=0$$

Combine 98\left(x+8-6\sqrt{x-7}\right) and -196\left(x+8-6\sqrt{x-7}\right) to get -98\left(x+8-6\sqrt{x-7}\right).

$$-2\left(x+3-4\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply -2 by x+3-4\sqrt{x-7}.

$$\left(-2x-6+8\sqrt{x-7}\right)\left(x+8-6\sqrt{x-7}\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply -2x-6+8\sqrt{x-7} by x+8-6\sqrt{x-7} and combine like terms.

$$-2x^{2}-22x+20x\sqrt{x-7}-48+100\sqrt{x-7}-48\left(\sqrt{x-7}\right)^{2}+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Calculate \sqrt{x-7} to the power of 2 and get x-7.

$$-2x^{2}-22x+20x\sqrt{x-7}-48+100\sqrt{x-7}-48\left(x-7\right)+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply -48 by x-7.

$$-2x^{2}-22x+20x\sqrt{x-7}-48+100\sqrt{x-7}-48x+336+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine -22x and -48x to get -70x.

$$-2x^{2}-70x+20x\sqrt{x-7}-48+100\sqrt{x-7}+336+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Add -48 and 336 to get 288.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+\left(x+3-4\sqrt{x-7}\right)^{2}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Square x+3-4\sqrt{x-7}.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+x^{2}-8\sqrt{x-7}x+16\left(\sqrt{x-7}\right)^{2}+6x-24\sqrt{x-7}+9+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Calculate \sqrt{x-7} to the power of 2 and get x-7.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+x^{2}-8\sqrt{x-7}x+16\left(x-7\right)+6x-24\sqrt{x-7}+9+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply 16 by x-7.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+x^{2}-8\sqrt{x-7}x+16x-112+6x-24\sqrt{x-7}+9+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 16x and 6x to get 22x.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+x^{2}-8\sqrt{x-7}x+22x-112-24\sqrt{x-7}+9+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Add -112 and 9 to get -103.

$$-2x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}+x^{2}-8\sqrt{x-7}x+22x-103-24\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

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

$$-x^{2}-70x+20x\sqrt{x-7}+288+100\sqrt{x-7}-8\sqrt{x-7}x+22x-103-24\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 20x\sqrt{x-7} and -8\sqrt{x-7}x to get 12x\sqrt{x-7}.

$$-x^{2}-70x+12x\sqrt{x-7}+288+100\sqrt{x-7}+22x-103-24\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine -70x and 22x to get -48x.

$$-x^{2}-48x+12x\sqrt{x-7}+288+100\sqrt{x-7}-103-24\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Subtract 103 from 288 to get 185.

$$-x^{2}-48x+12x\sqrt{x-7}+185+100\sqrt{x-7}-24\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 100\sqrt{x-7} and -24\sqrt{x-7} to get 76\sqrt{x-7}.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+\left(x+8-6\sqrt{x-7}\right)^{2}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Square x+8-6\sqrt{x-7}.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+x^{2}-12\sqrt{x-7}x+36\left(\sqrt{x-7}\right)^{2}+16x-96\sqrt{x-7}+64-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Calculate \sqrt{x-7} to the power of 2 and get x-7.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+x^{2}-12\sqrt{x-7}x+36\left(x-7\right)+16x-96\sqrt{x-7}+64-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply 36 by x-7.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+x^{2}-12\sqrt{x-7}x+36x-252+16x-96\sqrt{x-7}+64-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 36x and 16x to get 52x.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+x^{2}-12\sqrt{x-7}x+52x-252-96\sqrt{x-7}+64-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Add -252 and 64 to get -188.

$$-x^{2}-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}+x^{2}-12\sqrt{x-7}x+52x-188-96\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine -x^{2} and x^{2} to get 0.

$$-48x+12x\sqrt{x-7}+185+76\sqrt{x-7}-12\sqrt{x-7}x+52x-188-96\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 12x\sqrt{x-7} and -12\sqrt{x-7}x to get 0.

$$-48x+185+76\sqrt{x-7}+52x-188-96\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine -48x and 52x to get 4x.

$$4x+185+76\sqrt{x-7}-188-96\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Subtract 188 from 185 to get -3.

$$4x-3+76\sqrt{x-7}-96\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 76\sqrt{x-7} and -96\sqrt{x-7} to get -20\sqrt{x-7}.

$$4x-3-20\sqrt{x-7}-98\left(x+3-4\sqrt{x-7}\right)-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply -98 by x+3-4\sqrt{x-7}.

$$4x-3-20\sqrt{x-7}-98x-294+392\sqrt{x-7}-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine 4x and -98x to get -94x.

$$-94x-3-20\sqrt{x-7}-294+392\sqrt{x-7}-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Subtract 294 from -3 to get -297.

$$-94x-297-20\sqrt{x-7}+392\sqrt{x-7}-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Combine -20\sqrt{x-7} and 392\sqrt{x-7} to get 372\sqrt{x-7}.

$$-94x-297+372\sqrt{x-7}-98\left(x+8-6\sqrt{x-7}\right)+2401=0$$

Use the distributive property to multiply -98 by x+8-6\sqrt{x-7}.

$$-94x-297+372\sqrt{x-7}-98x-784+588\sqrt{x-7}+2401=0$$

Combine -94x and -98x to get -192x.

$$-192x-297+372\sqrt{x-7}-784+588\sqrt{x-7}+2401=0$$

Subtract 784 from -297 to get -1081.

$$-192x-1081+372\sqrt{x-7}+588\sqrt{x-7}+2401=0$$

Combine 372\sqrt{x-7} and 588\sqrt{x-7} to get 960\sqrt{x-7}.

$$-192x-1081+960\sqrt{x-7}+2401=0$$

Add -1081 and 2401 to get 1320.

$$-192x+1320+960\sqrt{x-7}=0$$

Subtract 1320 from both sides. Anything subtracted from zero gives its negation.

$$-192x+960\sqrt{x-7}=-1320$$

Subtract -192x from both sides of the equation.

$$960\sqrt{x-7}=-1320+192x$$

Square both sides of the equation.

$$\left(960\sqrt{x-7}\right)^{2}=\left(192x-1320\right)^{2}$$

Expand \left(960\sqrt{x-7}\right)^{2}.

$$960^{2}\left(\sqrt{x-7}\right)^{2}=\left(192x-1320\right)^{2}$$

Calculate 960 to the power of 2 and get 921600.

$$921600\left(\sqrt{x-7}\right)^{2}=\left(192x-1320\right)^{2}$$

Calculate \sqrt{x-7} to the power of 2 and get x-7.

$$921600\left(x-7\right)=\left(192x-1320\right)^{2}$$

Use the distributive property to multiply 921600 by x-7.

$$921600x-6451200=\left(192x-1320\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(192x-1320\right)^{2}.

$$921600x-6451200=36864x^{2}-506880x+1742400$$

Subtract 36864x^{2} from both sides.

$$921600x-6451200-36864x^{2}=-506880x+1742400$$

Add 506880x to both sides.

$$921600x-6451200-36864x^{2}+506880x=1742400$$

Combine 921600x and 506880x to get 1428480x.

$$1428480x-6451200-36864x^{2}=1742400$$

Subtract 1742400 from both sides.

$$1428480x-6451200-36864x^{2}-1742400=0$$

Subtract 1742400 from -6451200 to get -8193600.

$$1428480x-8193600-36864x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-36864x^{2}+1428480x-8193600=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -36864 for a, 1428480 for b, and -8193600 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1428480±\sqrt{1428480^{2}-4\left(-36864\right)\left(-8193600\right)}}{2\left(-36864\right)}$$

Square 1428480.

$$x=\frac{-1428480±\sqrt{2040555110400-4\left(-36864\right)\left(-8193600\right)}}{2\left(-36864\right)}$$

Multiply -4 times -36864.

$$x=\frac{-1428480±\sqrt{2040555110400+147456\left(-8193600\right)}}{2\left(-36864\right)}$$

Multiply 147456 times -8193600.

$$x=\frac{-1428480±\sqrt{2040555110400-1208195481600}}{2\left(-36864\right)}$$

Add 2040555110400 to -1208195481600.

$$x=\frac{-1428480±\sqrt{832359628800}}{2\left(-36864\right)}$$

Take the square root of 832359628800.

$$x=\frac{-1428480±645120\sqrt{2}}{2\left(-36864\right)}$$

Multiply 2 times -36864.

$$x=\frac{-1428480±645120\sqrt{2}}{-73728}$$

Now solve the equation x=\frac{-1428480±645120\sqrt{2}}{-73728} when ± is plus. Add -1428480 to 645120\sqrt{2}.

$$x=\frac{645120\sqrt{2}-1428480}{-73728}$$

Divide -1428480+645120\sqrt{2} by -73728.

$$x=-\frac{35\sqrt{2}}{4}+\frac{155}{8}$$

Now solve the equation x=\frac{-1428480±645120\sqrt{2}}{-73728} when ± is minus. Subtract 645120\sqrt{2} from -1428480.

$$x=\frac{-645120\sqrt{2}-1428480}{-73728}$$

Divide -1428480-645120\sqrt{2} by -73728.

$$x=\frac{35\sqrt{2}}{4}+\frac{155}{8}$$

The equation is now solved.

$$x=-\frac{35\sqrt{2}}{4}+\frac{155}{8} x=\frac{35\sqrt{2}}{4}+\frac{155}{8}$$

Substitute -\frac{35\sqrt{2}}{4}+\frac{155}{8} for x in the equation \sqrt{x+3-4\sqrt{x-7}}+\sqrt{x+8-6\sqrt{x-7}}=7.

$$\sqrt{-\frac{35\sqrt{2}}{4}+\frac{155}{8}+3-4\sqrt{-\frac{35\sqrt{2}}{4}+\frac{155}{8}-7}}+\sqrt{-\frac{35\sqrt{2}}{4}+\frac{155}{8}+8-6\sqrt{-\frac{35\sqrt{2}}{4}+\frac{155}{8}-7}}=7$$

Simplify. The value x=-\frac{35\sqrt{2}}{4}+\frac{155}{8} satisfies the equation.

$$7=7$$

Substitute \frac{35\sqrt{2}}{4}+\frac{155}{8} for x in the equation \sqrt{x+3-4\sqrt{x-7}}+\sqrt{x+8-6\sqrt{x-7}}=7.

$$\sqrt{\frac{35\sqrt{2}}{4}+\frac{155}{8}+3-4\sqrt{\frac{35\sqrt{2}}{4}+\frac{155}{8}-7}}+\sqrt{\frac{35\sqrt{2}}{4}+\frac{155}{8}+8-6\sqrt{\frac{35\sqrt{2}}{4}+\frac{155}{8}-7}}=7$$

Simplify. The value x=\frac{35\sqrt{2}}{4}+\frac{155}{8} satisfies the equation.

$$7=7$$

List all solutions of \sqrt{x+3-4\sqrt{x-7}}=-\sqrt{x+8-6\sqrt{x-7}}+7.

$$x=-\frac{35\sqrt{2}}{4}+\frac{155}{8} x=\frac{35\sqrt{2}}{4}+\frac{155}{8}$$