Solve sqrt{184-56x+2x^2}+sqrt{441-42x+2x^2}=35

$$\sqrt { 184 - 56 x + 2 x ^ { 2 } } + \sqrt { 441 - 42 x + 2 x ^ { 2 } } = 35$$

Solve for x

$$x=\frac{4416-150\sqrt{886}}{343}\approx -0.142457201$$

$$x = \frac{150 \sqrt{886} + 4416}{343} \approx 25.891728338$$

Solution Steps

Subtract \sqrt{441-42x+2x^{2}} from both sides of the equation.

$$\sqrt{184-56x+2x^{2}}=35-\sqrt{441-42x+2x^{2}}$$

Square both sides of the equation.

$$\left(\sqrt{184-56x+2x^{2}}\right)^{2}=\left(35-\sqrt{441-42x+2x^{2}}\right)^{2}$$

Calculate \sqrt{184-56x+2x^{2}} to the power of 2 and get 184-56x+2x^{2}.

$$184-56x+2x^{2}=\left(35-\sqrt{441-42x+2x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(35-\sqrt{441-42x+2x^{2}}\right)^{2}.

$$184-56x+2x^{2}=1225-70\sqrt{441-42x+2x^{2}}+\left(\sqrt{441-42x+2x^{2}}\right)^{2}$$

Calculate \sqrt{441-42x+2x^{2}} to the power of 2 and get 441-42x+2x^{2}.

$$184-56x+2x^{2}=1225-70\sqrt{441-42x+2x^{2}}+441-42x+2x^{2}$$

Add 1225 and 441 to get 1666.

$$184-56x+2x^{2}=1666-70\sqrt{441-42x+2x^{2}}-42x+2x^{2}$$

Subtract 1666-42x+2x^{2} from both sides of the equation.

$$184-56x+2x^{2}-\left(1666-42x+2x^{2}\right)=-70\sqrt{441-42x+2x^{2}}$$

To find the opposite of 1666-42x+2x^{2}, find the opposite of each term.

$$184-56x+2x^{2}-1666+42x-2x^{2}=-70\sqrt{441-42x+2x^{2}}$$

Subtract 1666 from 184 to get -1482.

$$-1482-56x+2x^{2}+42x-2x^{2}=-70\sqrt{441-42x+2x^{2}}$$

Combine -56x and 42x to get -14x.

$$-1482-14x+2x^{2}-2x^{2}=-70\sqrt{441-42x+2x^{2}}$$

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

$$-1482-14x=-70\sqrt{441-42x+2x^{2}}$$

Square both sides of the equation.

$$\left(-1482-14x\right)^{2}=\left(-70\sqrt{441-42x+2x^{2}}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-1482-14x\right)^{2}.

$$2196324+41496x+196x^{2}=\left(-70\sqrt{441-42x+2x^{2}}\right)^{2}$$

Expand \left(-70\sqrt{441-42x+2x^{2}}\right)^{2}.

$$2196324+41496x+196x^{2}=\left(-70\right)^{2}\left(\sqrt{441-42x+2x^{2}}\right)^{2}$$

Calculate -70 to the power of 2 and get 4900.

$$2196324+41496x+196x^{2}=4900\left(\sqrt{441-42x+2x^{2}}\right)^{2}$$

Calculate \sqrt{441-42x+2x^{2}} to the power of 2 and get 441-42x+2x^{2}.

$$2196324+41496x+196x^{2}=4900\left(441-42x+2x^{2}\right)$$

Use the distributive property to multiply 4900 by 441-42x+2x^{2}.

$$2196324+41496x+196x^{2}=2160900-205800x+9800x^{2}$$

Subtract 2160900 from both sides.

$$2196324+41496x+196x^{2}-2160900=-205800x+9800x^{2}$$

Subtract 2160900 from 2196324 to get 35424.

$$35424+41496x+196x^{2}=-205800x+9800x^{2}$$

Add 205800x to both sides.

$$35424+41496x+196x^{2}+205800x=9800x^{2}$$

Combine 41496x and 205800x to get 247296x.

$$35424+247296x+196x^{2}=9800x^{2}$$

Subtract 9800x^{2} from both sides.

$$35424+247296x+196x^{2}-9800x^{2}=0$$

Combine 196x^{2} and -9800x^{2} to get -9604x^{2}.

$$35424+247296x-9604x^{2}=0$$

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.

$$-9604x^{2}+247296x+35424=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -9604 for a, 247296 for b, and 35424 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-247296±\sqrt{247296^{2}-4\left(-9604\right)\times 35424}}{2\left(-9604\right)}$$

Square 247296.

$$x=\frac{-247296±\sqrt{61155311616-4\left(-9604\right)\times 35424}}{2\left(-9604\right)}$$

Multiply -4 times -9604.

$$x=\frac{-247296±\sqrt{61155311616+38416\times 35424}}{2\left(-9604\right)}$$

Multiply 38416 times 35424.

$$x=\frac{-247296±\sqrt{61155311616+1360848384}}{2\left(-9604\right)}$$

Add 61155311616 to 1360848384.

$$x=\frac{-247296±\sqrt{62516160000}}{2\left(-9604\right)}$$

Take the square root of 62516160000.

$$x=\frac{-247296±8400\sqrt{886}}{2\left(-9604\right)}$$

Multiply 2 times -9604.

$$x=\frac{-247296±8400\sqrt{886}}{-19208}$$

Now solve the equation x=\frac{-247296±8400\sqrt{886}}{-19208} when ± is plus. Add -247296 to 8400\sqrt{886}.

$$x=\frac{8400\sqrt{886}-247296}{-19208}$$

Divide -247296+8400\sqrt{886} by -19208.

$$x=\frac{4416-150\sqrt{886}}{343}$$

Now solve the equation x=\frac{-247296±8400\sqrt{886}}{-19208} when ± is minus. Subtract 8400\sqrt{886} from -247296.

$$x=\frac{-8400\sqrt{886}-247296}{-19208}$$

Divide -247296-8400\sqrt{886} by -19208.

$$x=\frac{150\sqrt{886}+4416}{343}$$

The equation is now solved.

$$x=\frac{4416-150\sqrt{886}}{343} x=\frac{150\sqrt{886}+4416}{343}$$

Substitute \frac{4416-150\sqrt{886}}{343} for x in the equation \sqrt{184-56x+2x^{2}}+\sqrt{441-42x+2x^{2}}=35.

$$\sqrt{184-56\times \frac{4416-150\sqrt{886}}{343}+2\times \left(\frac{4416-150\sqrt{886}}{343}\right)^{2}}+\sqrt{441-42\times \frac{4416-150\sqrt{886}}{343}+2\times \left(\frac{4416-150\sqrt{886}}{343}\right)^{2}}=35$$

Simplify. The value x=\frac{4416-150\sqrt{886}}{343} satisfies the equation.

$$\left(\frac{15697000}{117649}+\frac{231600}{117649}\times 886^{\frac{1}{2}}\right)^{\frac{1}{2}}+\frac{8145}{343}-\frac{30}{343}\times 886^{\frac{1}{2}}=35$$

Substitute \frac{150\sqrt{886}+4416}{343} for x in the equation \sqrt{184-56x+2x^{2}}+\sqrt{441-42x+2x^{2}}=35.

$$\sqrt{184-56\times \frac{150\sqrt{886}+4416}{343}+2\times \left(\frac{150\sqrt{886}+4416}{343}\right)^{2}}+\sqrt{441-42\times \frac{150\sqrt{886}+4416}{343}+2\times \left(\frac{150\sqrt{886}+4416}{343}\right)^{2}}=35$$

Simplify. The value x=\frac{150\sqrt{886}+4416}{343} satisfies the equation.

$$\left(\frac{15697000}{117649}-\frac{231600}{117649}\times 886^{\frac{1}{2}}\right)^{\frac{1}{2}}+\frac{8145}{343}+\frac{30}{343}\times 886^{\frac{1}{2}}=35$$

List all solutions of \sqrt{2x^{2}-56x+184}=-\sqrt{2x^{2}-42x+441}+35.

$$x=\frac{4416-150\sqrt{886}}{343} x=\frac{150\sqrt{886}+4416}{343}$$