Solve x-4=sqrt{5x-14}

$$x - 4 = \sqrt { 5 x - 14 }$$

Solve for x

$$x=10$$

Solution Steps

Square both sides of the equation.

$$\left(x-4\right)^{2}=\left(\sqrt{5x-14}\right)^{2}$$

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

$$x^{2}-8x+16=\left(\sqrt{5x-14}\right)^{2}$$

Calculate \sqrt{5x-14} to the power of 2 and get 5x-14.

$$x^{2}-8x+16=5x-14$$

Subtract 5x from both sides.

$$x^{2}-8x+16-5x=-14$$

Combine -8x and -5x to get -13x.

$$x^{2}-13x+16=-14$$

Add 14 to both sides.

$$x^{2}-13x+16+14=0$$

Add 16 and 14 to get 30.

$$x^{2}-13x+30=0$$

To solve the equation, factor x^{2}-13x+30 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.

$$a+b=-13 ab=30$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 30.

$$-1,-30 -2,-15 -3,-10 -5,-6$$

Calculate the sum for each pair.

$$-1-30=-31 -2-15=-17 -3-10=-13 -5-6=-11$$

The solution is the pair that gives sum -13.

$$a=-10 b=-3$$

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

$$\left(x-10\right)\left(x-3\right)$$

To find equation solutions, solve x-10=0 and x-3=0.

$$x=10 x=3$$

Substitute 10 for x in the equation x-4=\sqrt{5x-14}.

$$10-4=\sqrt{5\times 10-14}$$

Simplify. The value x=10 satisfies the equation.

$$6=6$$

Substitute 3 for x in the equation x-4=\sqrt{5x-14}.

$$3-4=\sqrt{5\times 3-14}$$

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

$$-1=1$$

Equation x-4=\sqrt{5x-14} has a unique solution.

$$x=10$$