Solve n=sqrt{-25+10n}

$$n = \sqrt { - 25 + 10 n }$$

Solve for n

$$n=5$$

Solution Steps

Square both sides of the equation.

$$n^{2}=\left(\sqrt{-25+10n}\right)^{2}$$

Calculate \sqrt{-25+10n} to the power of 2 and get -25+10n.

$$n^{2}=-25+10n$$

Subtract -25 from both sides.

$$n^{2}-\left(-25\right)=10n$$

The opposite of -25 is 25.

$$n^{2}+25=10n$$

Subtract 10n from both sides.

$$n^{2}+25-10n=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$n^{2}-10n+25=0$$

To solve the equation, factor n^{2}-10n+25 using formula n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). To find a and b, set up a system to be solved.

$$a+b=-10 ab=25$$

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 25.

$$-1,-25 -5,-5$$

Calculate the sum for each pair.

$$-1-25=-26 -5-5=-10$$

The solution is the pair that gives sum -10.

$$a=-5 b=-5$$

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

$$\left(n-5\right)\left(n-5\right)$$

Rewrite as a binomial square.

$$\left(n-5\right)^{2}$$

To find equation solution, solve n-5=0.

$$n=5$$

Substitute 5 for n in the equation n=\sqrt{-25+10n}.

$$5=\sqrt{-25+10\times 5}$$

Simplify. The value n=5 satisfies the equation.

$$5=5$$

Equation n=\sqrt{10n-25} has a unique solution.

$$n=5$$