Solve x-4=sqrt{-5x+34}

$$x - 4 = \sqrt { - 5 x + 34 }$$

Solve for x

$$x=6$$

Solution Steps

Square both sides of the equation.

$$\left(x-4\right)^{2}=\left(\sqrt{-5x+34}\right)^{2}$$

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

$$x^{2}-8x+16=\left(\sqrt{-5x+34}\right)^{2}$$

Calculate \sqrt{-5x+34} to the power of 2 and get -5x+34.

$$x^{2}-8x+16=-5x+34$$

Add 5x to both sides.

$$x^{2}-8x+16+5x=34$$

Combine -8x and 5x to get -3x.

$$x^{2}-3x+16=34$$

Subtract 34 from both sides.

$$x^{2}-3x+16-34=0$$

Subtract 34 from 16 to get -18.

$$x^{2}-3x-18=0$$

To solve the equation, factor x^{2}-3x-18 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-3 ab=-18$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -18.

$$1,-18 2,-9 3,-6$$

Calculate the sum for each pair.

$$1-18=-17 2-9=-7 3-6=-3$$

The solution is the pair that gives sum -3.

$$a=-6 b=3$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-6\right)\left(x+3\right)$$

To find equation solutions, solve x-6=0 and x+3=0.

$$x=6 x=-3$$

Substitute 6 for x in the equation x-4=\sqrt{-5x+34}.

$$6-4=\sqrt{-5\times 6+34}$$

Simplify. The value x=6 satisfies the equation.

$$2=2$$

Substitute -3 for x in the equation x-4=\sqrt{-5x+34}.

$$-3-4=\sqrt{-5\left(-3\right)+34}$$

Simplify. The value x=-3 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-7=7$$

Equation x-4=\sqrt{34-5x} has a unique solution.

$$x=6$$