Solve sqrt{-45-14x}=-x.Ec

$$\sqrt { - 45 - 14 x } = - x . E c$$

Solve for x

$$x=-5$$

$$x=-9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{-45-14x}\right)^{2}=\left(-x\right)^{2}$$

Calculate \sqrt{-45-14x} to the power of 2 and get -45-14x.

$$-45-14x=\left(-x\right)^{2}$$

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

$$-45-14x=x^{2}$$

Subtract x^{2} from both sides.

$$-45-14x-x^{2}=0$$

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

$$-x^{2}-14x-45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-45. To find a and b, set up a system to be solved.

$$a+b=-14 ab=-\left(-45\right)=45$$

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 45.

$$-1,-45 -3,-15 -5,-9$$

Calculate the sum for each pair.

$$-1-45=-46 -3-15=-18 -5-9=-14$$

The solution is the pair that gives sum -14.

$$a=-5 b=-9$$

Rewrite -x^{2}-14x-45 as \left(-x^{2}-5x\right)+\left(-9x-45\right).

$$\left(-x^{2}-5x\right)+\left(-9x-45\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(-x-5\right)+9\left(-x-5\right)$$

Factor out common term -x-5 by using distributive property.

$$\left(-x-5\right)\left(x+9\right)$$

To find equation solutions, solve -x-5=0 and x+9=0.

$$x=-5 x=-9$$

Substitute -5 for x in the equation \sqrt{-45-14x}=-x.

$$\sqrt{-45-14\left(-5\right)}=-\left(-5\right)$$

Simplify. The value x=-5 satisfies the equation.

$$5=5$$

Substitute -9 for x in the equation \sqrt{-45-14x}=-x.

$$\sqrt{-45-14\left(-9\right)}=-\left(-9\right)$$

Simplify. The value x=-9 satisfies the equation.

$$9=9$$

List all solutions of \sqrt{-14x-45}=-x.

$$x=-5 x=-9$$