Solve sqrt{-5n+14}=-n

$$\sqrt { - 5 n + 14 } = - n$$

Solve for n

$$n=-7$$

Solution Steps

Square both sides of the equation.

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

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

$$-5n+14=\left(-n\right)^{2}$$

Calculate -n to the power of 2 and get n^{2}.

$$-5n+14=n^{2}$$

Subtract n^{2} from both sides.

$$-5n+14-n^{2}=0$$

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

$$-n^{2}-5n+14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn+14. To find a and b, set up a system to be solved.

$$a+b=-5 ab=-14=-14$$

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 -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -5.

$$a=2 b=-7$$

Rewrite -n^{2}-5n+14 as \left(-n^{2}+2n\right)+\left(-7n+14\right).

$$\left(-n^{2}+2n\right)+\left(-7n+14\right)$$

Factor out n in the first and 7 in the second group.

$$n\left(-n+2\right)+7\left(-n+2\right)$$

Factor out common term -n+2 by using distributive property.

$$\left(-n+2\right)\left(n+7\right)$$

To find equation solutions, solve -n+2=0 and n+7=0.

$$n=2 n=-7$$

Substitute 2 for n in the equation \sqrt{-5n+14}=-n.

$$\sqrt{-5\times 2+14}=-2$$

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

$$2=-2$$

Substitute -7 for n in the equation \sqrt{-5n+14}=-n.

$$\sqrt{-5\left(-7\right)+14}=-\left(-7\right)$$

Simplify. The value n=-7 satisfies the equation.

$$7=7$$

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

$$n=-7$$