Solve (-x+7)(sqrt{x^2+2x+5})=(x+1)(sqrt{x^2-14x+65})

$$( - x + 7 ) ( \sqrt { x ^ { 2 } + 2 x + 5 } ) = ( x + 1 ) ( \sqrt { x ^ { 2 } - 14 x + 65 } )$$

Solve for x

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Solution Steps

Square both sides of the equation.

$$\left(\left(-x+7\right)\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply -x+7 by \sqrt{x^{2}+2x+5}.

$$\left(\left(-x\right)\sqrt{x^{2}+2x+5}+7\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\left(-x\right)\sqrt{x^{2}+2x+5}+7\sqrt{x^{2}+2x+5}\right)^{2}.

$$\left(-x\right)^{2}\left(\sqrt{x^{2}+2x+5}\right)^{2}+14\left(-x\right)\sqrt{x^{2}+2x+5}\sqrt{x^{2}+2x+5}+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Multiply \sqrt{x^{2}+2x+5} and \sqrt{x^{2}+2x+5} to get \left(\sqrt{x^{2}+2x+5}\right)^{2}.

$$\left(-x\right)^{2}\left(\sqrt{x^{2}+2x+5}\right)^{2}+14\left(-x\right)\left(\sqrt{x^{2}+2x+5}\right)^{2}+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate -x to the power of 2 and get x^{2}.

$$x^{2}\left(\sqrt{x^{2}+2x+5}\right)^{2}+14\left(-x\right)\left(\sqrt{x^{2}+2x+5}\right)^{2}+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}+2x+5} to the power of 2 and get x^{2}+2x+5.

$$x^{2}\left(x^{2}+2x+5\right)+14\left(-x\right)\left(\sqrt{x^{2}+2x+5}\right)^{2}+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply x^{2} by x^{2}+2x+5.

$$x^{4}+2x^{3}+5x^{2}+14\left(-x\right)\left(\sqrt{x^{2}+2x+5}\right)^{2}+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}+2x+5} to the power of 2 and get x^{2}+2x+5.

$$x^{4}+2x^{3}+5x^{2}+14\left(-x\right)\left(x^{2}+2x+5\right)+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply 14\left(-x\right) by x^{2}+2x+5.

$$x^{4}+2x^{3}+5x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+49\left(\sqrt{x^{2}+2x+5}\right)^{2}=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}+2x+5} to the power of 2 and get x^{2}+2x+5.

$$x^{4}+2x^{3}+5x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+49\left(x^{2}+2x+5\right)=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply 49 by x^{2}+2x+5.

$$x^{4}+2x^{3}+5x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+49x^{2}+98x+245=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Combine 5x^{2} and 49x^{2} to get 54x^{2}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=\left(\left(x+1\right)\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply x+1 by \sqrt{x^{2}-14x+65}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=\left(x\sqrt{x^{2}-14x+65}+\sqrt{x^{2}-14x+65}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x\sqrt{x^{2}-14x+65}+\sqrt{x^{2}-14x+65}\right)^{2}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{2}\left(\sqrt{x^{2}-14x+65}\right)^{2}+2x\sqrt{x^{2}-14x+65}\sqrt{x^{2}-14x+65}+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Multiply \sqrt{x^{2}-14x+65} and \sqrt{x^{2}-14x+65} to get \left(\sqrt{x^{2}-14x+65}\right)^{2}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{2}\left(\sqrt{x^{2}-14x+65}\right)^{2}+2x\left(\sqrt{x^{2}-14x+65}\right)^{2}+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}-14x+65} to the power of 2 and get x^{2}-14x+65.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{2}\left(x^{2}-14x+65\right)+2x\left(\sqrt{x^{2}-14x+65}\right)^{2}+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply x^{2} by x^{2}-14x+65.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-14x^{3}+65x^{2}+2x\left(\sqrt{x^{2}-14x+65}\right)^{2}+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}-14x+65} to the power of 2 and get x^{2}-14x+65.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-14x^{3}+65x^{2}+2x\left(x^{2}-14x+65\right)+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Use the distributive property to multiply 2x by x^{2}-14x+65.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-14x^{3}+65x^{2}+2x^{3}-28x^{2}+130x+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Combine -14x^{3} and 2x^{3} to get -12x^{3}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-12x^{3}+65x^{2}-28x^{2}+130x+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Combine 65x^{2} and -28x^{2} to get 37x^{2}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-12x^{3}+37x^{2}+130x+\left(\sqrt{x^{2}-14x+65}\right)^{2}$$

Calculate \sqrt{x^{2}-14x+65} to the power of 2 and get x^{2}-14x+65.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-12x^{3}+37x^{2}+130x+x^{2}-14x+65$$

Combine 37x^{2} and x^{2} to get 38x^{2}.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-12x^{3}+38x^{2}+130x-14x+65$$

Combine 130x and -14x to get 116x.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=x^{4}-12x^{3}+38x^{2}+116x+65$$

Subtract x^{4} from both sides.

$$x^{4}+2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245-x^{4}=-12x^{3}+38x^{2}+116x+65$$

Combine x^{4} and -x^{4} to get 0.

$$2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=-12x^{3}+38x^{2}+116x+65$$

Add 12x^{3} to both sides.

$$2x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245+12x^{3}=38x^{2}+116x+65$$

Combine 2x^{3} and 12x^{3} to get 14x^{3}.

$$14x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=38x^{2}+116x+65$$

Subtract 38x^{2} from both sides.

$$14x^{3}+54x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245-38x^{2}=116x+65$$

Combine 54x^{2} and -38x^{2} to get 16x^{2}.

$$14x^{3}+16x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245=116x+65$$

Subtract 116x from both sides.

$$14x^{3}+16x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)+98x+245-116x=65$$

Combine 98x and -116x to get -18x.

$$14x^{3}+16x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)-18x+245=65$$

Subtract 65 from both sides.

$$14x^{3}+16x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)-18x+245-65=0$$

Subtract 65 from 245 to get 180.

$$14x^{3}+16x^{2}+14\left(-x\right)x^{2}+28\left(-x\right)x+70\left(-x\right)-18x+180=0$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$14x^{3}+16x^{2}+14\left(-1\right)x^{3}+28\left(-1\right)xx+70\left(-1\right)x-18x+180=0$$

Multiply 14 and -1 to get -14.

$$14x^{3}+16x^{2}-14x^{3}+28\left(-1\right)xx+70\left(-1\right)x-18x+180=0$$

Combine 14x^{3} and -14x^{3} to get 0.

$$16x^{2}+28\left(-1\right)xx+70\left(-1\right)x-18x+180=0$$

Multiply x and x to get x^{2}.

$$16x^{2}+28\left(-1\right)x^{2}+70\left(-1\right)x-18x+180=0$$

Multiply 28 and -1 to get -28.

$$16x^{2}-28x^{2}+70\left(-1\right)x-18x+180=0$$

Combine 16x^{2} and -28x^{2} to get -12x^{2}.

$$-12x^{2}+70\left(-1\right)x-18x+180=0$$

Multiply 70 and -1 to get -70.

$$-12x^{2}-70x-18x+180=0$$

Combine -70x and -18x to get -88x.

$$-12x^{2}-88x+180=0$$

Divide both sides by 4.

$$-3x^{2}-22x+45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3x^{2}+ax+bx+45. To find a and b, set up a system to be solved.

$$a+b=-22 ab=-3\times 45=-135$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -135.

$$1,-135 3,-45 5,-27 9,-15$$

Calculate the sum for each pair.

$$1-135=-134 3-45=-42 5-27=-22 9-15=-6$$

The solution is the pair that gives sum -22.

$$a=5 b=-27$$

Rewrite -3x^{2}-22x+45 as \left(-3x^{2}+5x\right)+\left(-27x+45\right).

$$\left(-3x^{2}+5x\right)+\left(-27x+45\right)$$

Factor out -x in the first and -9 in the second group.

$$-x\left(3x-5\right)-9\left(3x-5\right)$$

Factor out common term 3x-5 by using distributive property.

$$\left(3x-5\right)\left(-x-9\right)$$

To find equation solutions, solve 3x-5=0 and -x-9=0.

$$x=\frac{5}{3} x=-9$$

Substitute \frac{5}{3} for x in the equation \left(-x+7\right)\sqrt{x^{2}+2x+5}=\left(x+1\right)\sqrt{x^{2}-14x+65}.

$$\left(-\frac{5}{3}+7\right)\sqrt{\left(\frac{5}{3}\right)^{2}+2\times \frac{5}{3}+5}=\left(\frac{5}{3}+1\right)\sqrt{\left(\frac{5}{3}\right)^{2}-14\times \frac{5}{3}+65}$$

Simplify. The value x=\frac{5}{3} satisfies the equation.

$$\frac{160}{9}=\frac{160}{9}$$

Substitute -9 for x in the equation \left(-x+7\right)\sqrt{x^{2}+2x+5}=\left(x+1\right)\sqrt{x^{2}-14x+65}.

$$\left(-\left(-9\right)+7\right)\sqrt{\left(-9\right)^{2}+2\left(-9\right)+5}=\left(-9+1\right)\sqrt{\left(-9\right)^{2}-14\left(-9\right)+65}$$

Simplify. The value x=-9 does not satisfy the equation because the left and the right hand side have opposite signs.

$$32\times 17^{\frac{1}{2}}=-32\times 17^{\frac{1}{2}}$$

Equation \left(7-x\right)\sqrt{x^{2}+2x+5}=\left(x+1\right)\sqrt{x^{2}-14x+65} has a unique solution.

$$x=\frac{5}{3}$$