Solve 8+sqrt{2x}-3=3sqrt{x-1}

$$8 + \sqrt { 2 x } - 3 = 3 \sqrt { x - 1 }$$

Solve for x

$$x = \frac{578}{49} = 11\frac{39}{49} \approx 11.795918367$$

Solution Steps

Square both sides of the equation.

$$\left(8+\sqrt{2x}-3\right)^{2}=\left(3\sqrt{x-1}\right)^{2}$$

Subtract 3 from 8 to get 5.

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

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

$$25+10\sqrt{2x}+\left(\sqrt{2x}\right)^{2}=\left(3\sqrt{x-1}\right)^{2}$$

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

$$25+10\sqrt{2x}+2x=\left(3\sqrt{x-1}\right)^{2}$$

Expand \left(3\sqrt{x-1}\right)^{2}.

$$25+10\sqrt{2x}+2x=3^{2}\left(\sqrt{x-1}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$25+10\sqrt{2x}+2x=9\left(\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{x-1} to the power of 2 and get x-1.

$$25+10\sqrt{2x}+2x=9\left(x-1\right)$$

Use the distributive property to multiply 9 by x-1.

$$25+10\sqrt{2x}+2x=9x-9$$

Subtract 25+2x from both sides of the equation.

$$10\sqrt{2x}=9x-9-\left(25+2x\right)$$

To find the opposite of 25+2x, find the opposite of each term.

$$10\sqrt{2x}=9x-9-25-2x$$

Subtract 25 from -9 to get -34.

$$10\sqrt{2x}=9x-34-2x$$

Combine 9x and -2x to get 7x.

$$10\sqrt{2x}=7x-34$$

Square both sides of the equation.

$$\left(10\sqrt{2x}\right)^{2}=\left(7x-34\right)^{2}$$

Expand \left(10\sqrt{2x}\right)^{2}.

$$10^{2}\left(\sqrt{2x}\right)^{2}=\left(7x-34\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$100\left(\sqrt{2x}\right)^{2}=\left(7x-34\right)^{2}$$

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

$$100\times 2x=\left(7x-34\right)^{2}$$

Multiply 100 and 2 to get 200.

$$200x=\left(7x-34\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7x-34\right)^{2}.

$$200x=49x^{2}-476x+1156$$

Subtract 49x^{2} from both sides.

$$200x-49x^{2}=-476x+1156$$

Add 476x to both sides.

$$200x-49x^{2}+476x=1156$$

Combine 200x and 476x to get 676x.

$$676x-49x^{2}=1156$$

Subtract 1156 from both sides.

$$676x-49x^{2}-1156=0$$

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.

$$-49x^{2}+676x-1156=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 676 for b, and -1156 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-676±\sqrt{676^{2}-4\left(-49\right)\left(-1156\right)}}{2\left(-49\right)}$$

Square 676.

$$x=\frac{-676±\sqrt{456976-4\left(-49\right)\left(-1156\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-676±\sqrt{456976+196\left(-1156\right)}}{2\left(-49\right)}$$

Multiply 196 times -1156.

$$x=\frac{-676±\sqrt{456976-226576}}{2\left(-49\right)}$$

Add 456976 to -226576.

$$x=\frac{-676±\sqrt{230400}}{2\left(-49\right)}$$

Take the square root of 230400.

$$x=\frac{-676±480}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-676±480}{-98}$$

Now solve the equation x=\frac{-676±480}{-98} when ± is plus. Add -676 to 480.

$$x=-\frac{196}{-98}$$

Divide -196 by -98.

$$x=2$$

Now solve the equation x=\frac{-676±480}{-98} when ± is minus. Subtract 480 from -676.

$$x=-\frac{1156}{-98}$$

Reduce the fraction \frac{-1156}{-98} to lowest terms by extracting and canceling out 2.

$$x=\frac{578}{49}$$

The equation is now solved.

$$x=2 x=\frac{578}{49}$$

Substitute 2 for x in the equation 8+\sqrt{2x}-3=3\sqrt{x-1}.

$$8+\sqrt{2\times 2}-3=3\sqrt{2-1}$$

Simplify. The value x=2 does not satisfy the equation.

$$7=3$$

Substitute \frac{578}{49} for x in the equation 8+\sqrt{2x}-3=3\sqrt{x-1}.

$$8+\sqrt{2\times \frac{578}{49}}-3=3\sqrt{\frac{578}{49}-1}$$

Simplify. The value x=\frac{578}{49} satisfies the equation.

$$\frac{69}{7}=\frac{69}{7}$$

Equation \sqrt{2x}+5=3\sqrt{x-1} has a unique solution.

$$x=\frac{578}{49}$$