Solve x+1=sqrt{10x+21}

$$x+1= \sqrt{ 10x+21 }$$

Solve for x

$$x=10$$

Solution Steps

Square both sides of the equation.

$$\left(x+1\right)^{2}=\left(\sqrt{10x+21}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$x^{2}+2x+1=\left(\sqrt{10x+21}\right)^{2}$$

Calculate \sqrt{10x+21} to the power of 2 and get 10x+21.

$$x^{2}+2x+1=10x+21$$

Subtract 10x from both sides.

$$x^{2}+2x+1-10x=21$$

Combine 2x and -10x to get -8x.

$$x^{2}-8x+1=21$$

Subtract 21 from both sides.

$$x^{2}-8x+1-21=0$$

Subtract 21 from 1 to get -20.

$$x^{2}-8x-20=0$$

To solve the equation, factor x^{2}-8x-20 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.

$$a+b=-8 ab=-20$$

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 -20.

$$1,-20 2,-10 4,-5$$

Calculate the sum for each pair.

$$1-20=-19 2-10=-8 4-5=-1$$

The solution is the pair that gives sum -8.

$$a=-10 b=2$$

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

$$\left(x-10\right)\left(x+2\right)$$

To find equation solutions, solve x-10=0 and x+2=0.

$$x=10 x=-2$$

Substitute 10 for x in the equation x+1=\sqrt{10x+21}.

$$10+1=\sqrt{10\times 10+21}$$

Simplify. The value x=10 satisfies the equation.

$$11=11$$

Substitute -2 for x in the equation x+1=\sqrt{10x+21}.

$$-2+1=\sqrt{10\left(-2\right)+21}$$

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Equation x+1=\sqrt{10x+21} has a unique solution.

$$x=10$$