Solve -7sqrt{x-2}=x-4

$$- 7 \sqrt { x - 2 } = x - 4$$

Solve for x

$$x = \frac{57 - 7 \sqrt{57}}{2} \approx 2.075579477$$

Solution Steps

Square both sides of the equation.

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

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

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

Calculate -7 to the power of 2 and get 49.

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

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

$$49\left(x-2\right)=\left(x-4\right)^{2}$$

Use the distributive property to multiply 49 by x-2.

$$49x-98=\left(x-4\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-4\right)^{2}.

$$49x-98=x^{2}-8x+16$$

Subtract x^{2} from both sides.

$$49x-98-x^{2}=-8x+16$$

Add 8x to both sides.

$$49x-98-x^{2}+8x=16$$

Combine 49x and 8x to get 57x.

$$57x-98-x^{2}=16$$

Subtract 16 from both sides.

$$57x-98-x^{2}-16=0$$

Subtract 16 from -98 to get -114.

$$57x-114-x^{2}=0$$

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.

$$-x^{2}+57x-114=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 57 for b, and -114 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-57±\sqrt{57^{2}-4\left(-1\right)\left(-114\right)}}{2\left(-1\right)}$$

Square 57.

$$x=\frac{-57±\sqrt{3249-4\left(-1\right)\left(-114\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-57±\sqrt{3249+4\left(-114\right)}}{2\left(-1\right)}$$

Multiply 4 times -114.

$$x=\frac{-57±\sqrt{3249-456}}{2\left(-1\right)}$$

Add 3249 to -456.

$$x=\frac{-57±\sqrt{2793}}{2\left(-1\right)}$$

Take the square root of 2793.

$$x=\frac{-57±7\sqrt{57}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-57±7\sqrt{57}}{-2}$$

Now solve the equation x=\frac{-57±7\sqrt{57}}{-2} when ± is plus. Add -57 to 7\sqrt{57}.

$$x=\frac{7\sqrt{57}-57}{-2}$$

Divide -57+7\sqrt{57} by -2.

$$x=\frac{57-7\sqrt{57}}{2}$$

Now solve the equation x=\frac{-57±7\sqrt{57}}{-2} when ± is minus. Subtract 7\sqrt{57} from -57.

$$x=\frac{-7\sqrt{57}-57}{-2}$$

Divide -57-7\sqrt{57} by -2.

$$x=\frac{7\sqrt{57}+57}{2}$$

The equation is now solved.

$$x=\frac{57-7\sqrt{57}}{2} x=\frac{7\sqrt{57}+57}{2}$$

Substitute \frac{57-7\sqrt{57}}{2} for x in the equation -7\sqrt{x-2}=x-4.

$$-7\sqrt{\frac{57-7\sqrt{57}}{2}-2}=\frac{57-7\sqrt{57}}{2}-4$$

Simplify. The value x=\frac{57-7\sqrt{57}}{2} satisfies the equation.

$$\frac{49}{2}-\frac{7}{2}\times 57^{\frac{1}{2}}=\frac{49}{2}-\frac{7}{2}\times 57^{\frac{1}{2}}$$

Substitute \frac{7\sqrt{57}+57}{2} for x in the equation -7\sqrt{x-2}=x-4.

$$-7\sqrt{\frac{7\sqrt{57}+57}{2}-2}=\frac{7\sqrt{57}+57}{2}-4$$

Simplify. The value x=\frac{7\sqrt{57}+57}{2} does not satisfy the equation because the left and the right hand side have opposite signs.

$$-\frac{49}{2}-\frac{7}{2}\times 57^{\frac{1}{2}}=\frac{7}{2}\times 57^{\frac{1}{2}}+\frac{49}{2}$$

Equation -7\sqrt{x-2}=x-4 has a unique solution.

$$x=\frac{57-7\sqrt{57}}{2}$$