Solve x-2=sqrt{10x+19}

$$x - 2 = \sqrt { 10 x + 19 }$$

Solve for x

$$x=15$$

Solution Steps

Square both sides of the equation.

$$\left(x-2\right)^{2}=\left(\sqrt{10x+19}\right)^{2}$$

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

$$x^{2}-4x+4=\left(\sqrt{10x+19}\right)^{2}$$

Calculate \sqrt{10x+19} to the power of 2 and get 10x+19.

$$x^{2}-4x+4=10x+19$$

Subtract 10x from both sides.

$$x^{2}-4x+4-10x=19$$

Combine -4x and -10x to get -14x.

$$x^{2}-14x+4=19$$

Subtract 19 from both sides.

$$x^{2}-14x+4-19=0$$

Subtract 19 from 4 to get -15.

$$x^{2}-14x-15=0$$

To solve the equation, factor x^{2}-14x-15 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.

$$a+b=-14 ab=-15$$

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 -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -14.

$$a=-15 b=1$$

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

$$\left(x-15\right)\left(x+1\right)$$

To find equation solutions, solve x-15=0 and x+1=0.

$$x=15 x=-1$$

Substitute 15 for x in the equation x-2=\sqrt{10x+19}.

$$15-2=\sqrt{10\times 15+19}$$

Simplify. The value x=15 satisfies the equation.

$$13=13$$

Substitute -1 for x in the equation x-2=\sqrt{10x+19}.

$$-1-2=\sqrt{10\left(-1\right)+19}$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=3$$

Equation x-2=\sqrt{10x+19} has a unique solution.

$$x=15$$