Solve sqrt{x-6}=23x-56

$$\sqrt{ x-6 } =23x-56$$

Solve for x (complex solution)

$$x=\frac{2577+\sqrt{7543}i}{1058}\approx 2.435727788+0.082089269i$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-6}\right)^{2}=\left(23x-56\right)^{2}$$

Calculate \sqrt{x-6} to the power of 2 and get x-6.

$$x-6=\left(23x-56\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(23x-56\right)^{2}.

$$x-6=529x^{2}-2576x+3136$$

Subtract 529x^{2} from both sides.

$$x-6-529x^{2}=-2576x+3136$$

Add 2576x to both sides.

$$x-6-529x^{2}+2576x=3136$$

Combine x and 2576x to get 2577x.

$$2577x-6-529x^{2}=3136$$

Subtract 3136 from both sides.

$$2577x-6-529x^{2}-3136=0$$

Subtract 3136 from -6 to get -3142.

$$2577x-3142-529x^{2}=0$$

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.

$$-529x^{2}+2577x-3142=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -529 for a, 2577 for b, and -3142 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2577±\sqrt{2577^{2}-4\left(-529\right)\left(-3142\right)}}{2\left(-529\right)}$$

Square 2577.

$$x=\frac{-2577±\sqrt{6640929-4\left(-529\right)\left(-3142\right)}}{2\left(-529\right)}$$

Multiply -4 times -529.

$$x=\frac{-2577±\sqrt{6640929+2116\left(-3142\right)}}{2\left(-529\right)}$$

Multiply 2116 times -3142.

$$x=\frac{-2577±\sqrt{6640929-6648472}}{2\left(-529\right)}$$

Add 6640929 to -6648472.

$$x=\frac{-2577±\sqrt{-7543}}{2\left(-529\right)}$$

Take the square root of -7543.

$$x=\frac{-2577±\sqrt{7543}i}{2\left(-529\right)}$$

Multiply 2 times -529.

$$x=\frac{-2577±\sqrt{7543}i}{-1058}$$

Now solve the equation x=\frac{-2577±\sqrt{7543}i}{-1058} when ± is plus. Add -2577 to i\sqrt{7543}.

$$x=\frac{-2577+\sqrt{7543}i}{-1058}$$

Divide -2577+i\sqrt{7543} by -1058.

$$x=\frac{-\sqrt{7543}i+2577}{1058}$$

Now solve the equation x=\frac{-2577±\sqrt{7543}i}{-1058} when ± is minus. Subtract i\sqrt{7543} from -2577.

$$x=\frac{-\sqrt{7543}i-2577}{-1058}$$

Divide -2577-i\sqrt{7543} by -1058.

$$x=\frac{2577+\sqrt{7543}i}{1058}$$

The equation is now solved.

$$x=\frac{-\sqrt{7543}i+2577}{1058} x=\frac{2577+\sqrt{7543}i}{1058}$$

Substitute \frac{-\sqrt{7543}i+2577}{1058} for x in the equation \sqrt{x-6}=23x-56.

$$\sqrt{\frac{-\sqrt{7543}i+2577}{1058}-6}=23\times \frac{-\sqrt{7543}i+2577}{1058}-56$$

Simplify. The value x=\frac{-\sqrt{7543}i+2577}{1058} does not satisfy the equation.

$$-\left(\frac{1}{46}-\frac{1}{46}i\times 7543^{\frac{1}{2}}\right)=-\frac{1}{46}i\times 7543^{\frac{1}{2}}+\frac{1}{46}$$

Substitute \frac{2577+\sqrt{7543}i}{1058} for x in the equation \sqrt{x-6}=23x-56.

$$\sqrt{\frac{2577+\sqrt{7543}i}{1058}-6}=23\times \frac{2577+\sqrt{7543}i}{1058}-56$$

Simplify. The value x=\frac{2577+\sqrt{7543}i}{1058} satisfies the equation.

$$\frac{1}{46}+\frac{1}{46}i\times 7543^{\frac{1}{2}}=\frac{1}{46}+\frac{1}{46}i\times 7543^{\frac{1}{2}}$$

Equation \sqrt{x-6}=23x-56 has a unique solution.

$$x=\frac{2577+\sqrt{7543}i}{1058}$$