Solve 15-x=-sqrt{x-3}

$$15 - x = - \sqrt { x - 3 }$$

Solve for x

$$x=19$$

Solution Steps

Add \sqrt{x-3} to both sides.

$$15-x+\sqrt{x-3}=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$-x+\sqrt{x-3}=-15$$

Subtract -x from both sides of the equation.

$$\sqrt{x-3}=-15+x$$

Square both sides of the equation.

$$\left(\sqrt{x-3}\right)^{2}=\left(-15+x\right)^{2}$$

Calculate \sqrt{x-3} to the power of 2 and get x-3.

$$x-3=\left(-15+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-15+x\right)^{2}.

$$x-3=225-30x+x^{2}$$

Add 30x to both sides.

$$x-3+30x=225+x^{2}$$

Combine x and 30x to get 31x.

$$31x-3=225+x^{2}$$

Subtract x^{2} from both sides.

$$31x-3-x^{2}=225$$

Subtract 225 from both sides.

$$31x-3-x^{2}-225=0$$

Subtract 225 from -3 to get -228.

$$31x-228-x^{2}=0$$

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

$$-x^{2}+31x-228=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-228. To find a and b, set up a system to be solved.

$$a+b=31 ab=-\left(-228\right)=228$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 228.

$$1,228 2,114 3,76 4,57 6,38 12,19$$

Calculate the sum for each pair.

$$1+228=229 2+114=116 3+76=79 4+57=61 6+38=44 12+19=31$$

The solution is the pair that gives sum 31.

$$a=19 b=12$$

Rewrite -x^{2}+31x-228 as \left(-x^{2}+19x\right)+\left(12x-228\right).

$$\left(-x^{2}+19x\right)+\left(12x-228\right)$$

Factor out -x in the first and 12 in the second group.

$$-x\left(x-19\right)+12\left(x-19\right)$$

Factor out common term x-19 by using distributive property.

$$\left(x-19\right)\left(-x+12\right)$$

To find equation solutions, solve x-19=0 and -x+12=0.

$$x=19 x=12$$

Substitute 19 for x in the equation 15-x=-\sqrt{x-3}.

$$15-19=-\sqrt{19-3}$$

Simplify. The value x=19 satisfies the equation.

$$-4=-4$$

Substitute 12 for x in the equation 15-x=-\sqrt{x-3}.

$$15-12=-\sqrt{12-3}$$

Simplify. The value x=12 does not satisfy the equation because the left and the right hand side have opposite signs.

$$3=-3$$

Equation \sqrt{x-3}=x-15 has a unique solution.

$$x=19$$