Solve 10*8+sqrt{2x-3}=3sqrt{x-1}

$$10 \cdot 8 + \sqrt { 2 x - 3 } = 3 \sqrt { x - 1 }$$

Solve for x

$$x = \frac{480 \sqrt{12793} + 70442}{49} \approx 2545.570398408$$

Solution Steps

Square both sides of the equation.

$$\left(10\times 8+\sqrt{2x-3}\right)^{2}=\left(3\sqrt{x-1}\right)^{2}$$

Multiply 10 and 8 to get 80.

$$\left(80+\sqrt{2x-3}\right)^{2}=\left(3\sqrt{x-1}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(80+\sqrt{2x-3}\right)^{2}.

$$6400+160\sqrt{2x-3}+\left(\sqrt{2x-3}\right)^{2}=\left(3\sqrt{x-1}\right)^{2}$$

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

$$6400+160\sqrt{2x-3}+2x-3=\left(3\sqrt{x-1}\right)^{2}$$

Subtract 3 from 6400 to get 6397.

$$6397+160\sqrt{2x-3}+2x=\left(3\sqrt{x-1}\right)^{2}$$

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

$$6397+160\sqrt{2x-3}+2x=3^{2}\left(\sqrt{x-1}\right)^{2}$$

Calculate 3 to the power of 2 and get 9.

$$6397+160\sqrt{2x-3}+2x=9\left(\sqrt{x-1}\right)^{2}$$

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

$$6397+160\sqrt{2x-3}+2x=9\left(x-1\right)$$

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

$$6397+160\sqrt{2x-3}+2x=9x-9$$

Subtract 6397+2x from both sides of the equation.

$$160\sqrt{2x-3}=9x-9-\left(6397+2x\right)$$

To find the opposite of 6397+2x, find the opposite of each term.

$$160\sqrt{2x-3}=9x-9-6397-2x$$

Subtract 6397 from -9 to get -6406.

$$160\sqrt{2x-3}=9x-6406-2x$$

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

$$160\sqrt{2x-3}=7x-6406$$

Square both sides of the equation.

$$\left(160\sqrt{2x-3}\right)^{2}=\left(7x-6406\right)^{2}$$

Expand \left(160\sqrt{2x-3}\right)^{2}.

$$160^{2}\left(\sqrt{2x-3}\right)^{2}=\left(7x-6406\right)^{2}$$

Calculate 160 to the power of 2 and get 25600.

$$25600\left(\sqrt{2x-3}\right)^{2}=\left(7x-6406\right)^{2}$$

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

$$25600\left(2x-3\right)=\left(7x-6406\right)^{2}$$

Use the distributive property to multiply 25600 by 2x-3.

$$51200x-76800=\left(7x-6406\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7x-6406\right)^{2}.

$$51200x-76800=49x^{2}-89684x+41036836$$

Subtract 49x^{2} from both sides.

$$51200x-76800-49x^{2}=-89684x+41036836$$

Add 89684x to both sides.

$$51200x-76800-49x^{2}+89684x=41036836$$

Combine 51200x and 89684x to get 140884x.

$$140884x-76800-49x^{2}=41036836$$

Subtract 41036836 from both sides.

$$140884x-76800-49x^{2}-41036836=0$$

Subtract 41036836 from -76800 to get -41113636.

$$140884x-41113636-49x^{2}=0$$

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.

$$-49x^{2}+140884x-41113636=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -49 for a, 140884 for b, and -41113636 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-140884±\sqrt{140884^{2}-4\left(-49\right)\left(-41113636\right)}}{2\left(-49\right)}$$

Square 140884.

$$x=\frac{-140884±\sqrt{19848301456-4\left(-49\right)\left(-41113636\right)}}{2\left(-49\right)}$$

Multiply -4 times -49.

$$x=\frac{-140884±\sqrt{19848301456+196\left(-41113636\right)}}{2\left(-49\right)}$$

Multiply 196 times -41113636.

$$x=\frac{-140884±\sqrt{19848301456-8058272656}}{2\left(-49\right)}$$

Add 19848301456 to -8058272656.

$$x=\frac{-140884±\sqrt{11790028800}}{2\left(-49\right)}$$

Take the square root of 11790028800.

$$x=\frac{-140884±960\sqrt{12793}}{2\left(-49\right)}$$

Multiply 2 times -49.

$$x=\frac{-140884±960\sqrt{12793}}{-98}$$

Now solve the equation x=\frac{-140884±960\sqrt{12793}}{-98} when ± is plus. Add -140884 to 960\sqrt{12793}.

$$x=\frac{960\sqrt{12793}-140884}{-98}$$

Divide -140884+960\sqrt{12793} by -98.

$$x=\frac{70442-480\sqrt{12793}}{49}$$

Now solve the equation x=\frac{-140884±960\sqrt{12793}}{-98} when ± is minus. Subtract 960\sqrt{12793} from -140884.

$$x=\frac{-960\sqrt{12793}-140884}{-98}$$

Divide -140884-960\sqrt{12793} by -98.

$$x=\frac{480\sqrt{12793}+70442}{49}$$

The equation is now solved.

$$x=\frac{70442-480\sqrt{12793}}{49} x=\frac{480\sqrt{12793}+70442}{49}$$

Substitute \frac{70442-480\sqrt{12793}}{49} for x in the equation 10\times 8+\sqrt{2x-3}=3\sqrt{x-1}.

$$10\times 8+\sqrt{2\times \frac{70442-480\sqrt{12793}}{49}-3}=3\sqrt{\frac{70442-480\sqrt{12793}}{49}-1}$$

Simplify. The value x=\frac{70442-480\sqrt{12793}}{49} does not satisfy the equation.

$$\frac{400}{7}+\frac{3}{7}\times 12793^{\frac{1}{2}}=\frac{720}{7}-\frac{3}{7}\times 12793^{\frac{1}{2}}$$

Substitute \frac{480\sqrt{12793}+70442}{49} for x in the equation 10\times 8+\sqrt{2x-3}=3\sqrt{x-1}.

$$10\times 8+\sqrt{2\times \frac{480\sqrt{12793}+70442}{49}-3}=3\sqrt{\frac{480\sqrt{12793}+70442}{49}-1}$$

Simplify. The value x=\frac{480\sqrt{12793}+70442}{49} satisfies the equation.

$$\frac{720}{7}+\frac{3}{7}\times 12793^{\frac{1}{2}}=\frac{720}{7}+\frac{3}{7}\times 12793^{\frac{1}{2}}$$

Equation \sqrt{2x-3}+80=3\sqrt{x-1} has a unique solution.

$$x=\frac{480\sqrt{12793}+70442}{49}$$