Solve sqrt{x+7}+sqrt{x+2}=sqrt{18x}

$$\sqrt { x + 7 } + \sqrt { x + 2 } = \sqrt { 18 x }$$

Solve for x

$$x = \frac{\sqrt{554} + 27}{42} \approx 1.203266776$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+7}+\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$\left(\sqrt{x+7}\right)^{2}+2\sqrt{x+7}\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$x+7+2\sqrt{x+7}\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}=\left(\sqrt{18x}\right)^{2}$$

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

$$x+7+2\sqrt{x+7}\sqrt{x+2}+x+2=\left(\sqrt{18x}\right)^{2}$$

Combine x and x to get 2x.

$$2x+7+2\sqrt{x+7}\sqrt{x+2}+2=\left(\sqrt{18x}\right)^{2}$$

Add 7 and 2 to get 9.

$$2x+9+2\sqrt{x+7}\sqrt{x+2}=\left(\sqrt{18x}\right)^{2}$$

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

$$2x+9+2\sqrt{x+7}\sqrt{x+2}=18x$$

Subtract 2x+9 from both sides of the equation.

$$2\sqrt{x+7}\sqrt{x+2}=18x-\left(2x+9\right)$$

To find the opposite of 2x+9, find the opposite of each term.

$$2\sqrt{x+7}\sqrt{x+2}=18x-2x-9$$

Combine 18x and -2x to get 16x.

$$2\sqrt{x+7}\sqrt{x+2}=16x-9$$

Square both sides of the equation.

$$\left(2\sqrt{x+7}\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$2^{2}\left(\sqrt{x+7}\right)^{2}\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$4\left(\sqrt{x+7}\right)^{2}\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$4\left(x+7\right)\left(\sqrt{x+2}\right)^{2}=\left(16x-9\right)^{2}$$

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

$$4\left(x+7\right)\left(x+2\right)=\left(16x-9\right)^{2}$$

Use the distributive property to multiply 4 by x+7.

$$\left(4x+28\right)\left(x+2\right)=\left(16x-9\right)^{2}$$

Apply the distributive property by multiplying each term of 4x+28 by each term of x+2.

$$4x^{2}+8x+28x+56=\left(16x-9\right)^{2}$$

Combine 8x and 28x to get 36x.

$$4x^{2}+36x+56=\left(16x-9\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16x-9\right)^{2}.

$$4x^{2}+36x+56=256x^{2}-288x+81$$

Subtract 256x^{2} from both sides.

$$4x^{2}+36x+56-256x^{2}=-288x+81$$

Combine 4x^{2} and -256x^{2} to get -252x^{2}.

$$-252x^{2}+36x+56=-288x+81$$

Add 288x to both sides.

$$-252x^{2}+36x+56+288x=81$$

Combine 36x and 288x to get 324x.

$$-252x^{2}+324x+56=81$$

Subtract 81 from both sides.

$$-252x^{2}+324x+56-81=0$$

Subtract 81 from 56 to get -25.

$$-252x^{2}+324x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -252 for a, 324 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-324±\sqrt{324^{2}-4\left(-252\right)\left(-25\right)}}{2\left(-252\right)}$$

Square 324.

$$x=\frac{-324±\sqrt{104976-4\left(-252\right)\left(-25\right)}}{2\left(-252\right)}$$

Multiply -4 times -252.

$$x=\frac{-324±\sqrt{104976+1008\left(-25\right)}}{2\left(-252\right)}$$

Multiply 1008 times -25.

$$x=\frac{-324±\sqrt{104976-25200}}{2\left(-252\right)}$$

Add 104976 to -25200.

$$x=\frac{-324±\sqrt{79776}}{2\left(-252\right)}$$

Take the square root of 79776.

$$x=\frac{-324±12\sqrt{554}}{2\left(-252\right)}$$

Multiply 2 times -252.

$$x=\frac{-324±12\sqrt{554}}{-504}$$

Now solve the equation x=\frac{-324±12\sqrt{554}}{-504} when ± is plus. Add -324 to 12\sqrt{554}.

$$x=\frac{12\sqrt{554}-324}{-504}$$

Divide -324+12\sqrt{554} by -504.

$$x=-\frac{\sqrt{554}}{42}+\frac{9}{14}$$

Now solve the equation x=\frac{-324±12\sqrt{554}}{-504} when ± is minus. Subtract 12\sqrt{554} from -324.

$$x=\frac{-12\sqrt{554}-324}{-504}$$

Divide -324-12\sqrt{554} by -504.

$$x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$

The equation is now solved.

$$x=-\frac{\sqrt{554}}{42}+\frac{9}{14} x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$

Substitute -\frac{\sqrt{554}}{42}+\frac{9}{14} for x in the equation \sqrt{x+7}+\sqrt{x+2}=\sqrt{18x}.

$$\sqrt{-\frac{\sqrt{554}}{42}+\frac{9}{14}+7}+\sqrt{-\frac{\sqrt{554}}{42}+\frac{9}{14}+2}=\sqrt{18\left(-\frac{\sqrt{554}}{42}+\frac{9}{14}\right)}$$

Simplify. The value x=-\frac{\sqrt{554}}{42}+\frac{9}{14} does not satisfy the equation.

$$\left(-\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{107}{14}\right)^{\frac{1}{2}}+\left(-\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{37}{14}\right)^{\frac{1}{2}}=\left(-\frac{3}{7}\times 554^{\frac{1}{2}}+\frac{81}{7}\right)^{\frac{1}{2}}$$

Substitute \frac{\sqrt{554}}{42}+\frac{9}{14} for x in the equation \sqrt{x+7}+\sqrt{x+2}=\sqrt{18x}.

$$\sqrt{\frac{\sqrt{554}}{42}+\frac{9}{14}+7}+\sqrt{\frac{\sqrt{554}}{42}+\frac{9}{14}+2}=\sqrt{18\left(\frac{\sqrt{554}}{42}+\frac{9}{14}\right)}$$

Simplify. The value x=\frac{\sqrt{554}}{42}+\frac{9}{14} satisfies the equation.

$$\left(\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{107}{14}\right)^{\frac{1}{2}}+\left(\frac{1}{42}\times 554^{\frac{1}{2}}+\frac{37}{14}\right)^{\frac{1}{2}}=\left(\frac{3}{7}\times 554^{\frac{1}{2}}+\frac{81}{7}\right)^{\frac{1}{2}}$$

Equation \sqrt{x+2}+\sqrt{x+7}=\sqrt{18x} has a unique solution.

$$x=\frac{\sqrt{554}}{42}+\frac{9}{14}$$