Solve x+40=left(x-32right)^2

$$x+40= { \left(x-32 \right) }^{ 2 }$$

Solve for x

$$x=24$$

$$x=41$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-32\right)^{2}.

$$x+40=x^{2}-64x+1024$$

Subtract x^{2} from both sides.

$$x+40-x^{2}=-64x+1024$$

Add 64x to both sides.

$$x+40-x^{2}+64x=1024$$

Combine x and 64x to get 65x.

$$65x+40-x^{2}=1024$$

Subtract 1024 from both sides.

$$65x+40-x^{2}-1024=0$$

Subtract 1024 from 40 to get -984.

$$65x-984-x^{2}=0$$

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

$$-x^{2}+65x-984=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-984. To find a and b, set up a system to be solved.

$$a+b=65 ab=-\left(-984\right)=984$$

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 984.

$$1,984 2,492 3,328 4,246 6,164 8,123 12,82 24,41$$

Calculate the sum for each pair.

$$1+984=985 2+492=494 3+328=331 4+246=250 6+164=170 8+123=131 12+82=94 24+41=65$$

The solution is the pair that gives sum 65.

$$a=41 b=24$$

Rewrite -x^{2}+65x-984 as \left(-x^{2}+41x\right)+\left(24x-984\right).

$$\left(-x^{2}+41x\right)+\left(24x-984\right)$$

Factor out -x in the first and 24 in the second group.

$$-x\left(x-41\right)+24\left(x-41\right)$$

Factor out common term x-41 by using distributive property.

$$\left(x-41\right)\left(-x+24\right)$$

To find equation solutions, solve x-41=0 and -x+24=0.

$$x=41 x=24$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-32\right)^{2}.

$$x+40=x^{2}-64x+1024$$

Subtract x^{2} from both sides.

$$x+40-x^{2}=-64x+1024$$

Add 64x to both sides.

$$x+40-x^{2}+64x=1024$$

Combine x and 64x to get 65x.

$$65x+40-x^{2}=1024$$

Subtract 1024 from both sides.

$$65x+40-x^{2}-1024=0$$

Subtract 1024 from 40 to get -984.

$$65x-984-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+65x-984=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 65 for b, and -984 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65±\sqrt{65^{2}-4\left(-1\right)\left(-984\right)}}{2\left(-1\right)}$$

Square 65.

$$x=\frac{-65±\sqrt{4225-4\left(-1\right)\left(-984\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-65±\sqrt{4225+4\left(-984\right)}}{2\left(-1\right)}$$

Multiply 4 times -984.

$$x=\frac{-65±\sqrt{4225-3936}}{2\left(-1\right)}$$

Add 4225 to -3936.

$$x=\frac{-65±\sqrt{289}}{2\left(-1\right)}$$

Take the square root of 289.

$$x=\frac{-65±17}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-65±17}{-2}$$

Now solve the equation x=\frac{-65±17}{-2} when ± is plus. Add -65 to 17.

$$x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x=24$$

Now solve the equation x=\frac{-65±17}{-2} when ± is minus. Subtract 17 from -65.

$$x=-\frac{82}{-2}$$

Divide -82 by -2.

$$x=41$$

The equation is now solved.

$$x=24 x=41$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-32\right)^{2}.

$$x+40=x^{2}-64x+1024$$

Subtract x^{2} from both sides.

$$x+40-x^{2}=-64x+1024$$

Add 64x to both sides.

$$x+40-x^{2}+64x=1024$$

Combine x and 64x to get 65x.

$$65x+40-x^{2}=1024$$

Subtract 40 from both sides.

$$65x-x^{2}=1024-40$$

Subtract 40 from 1024 to get 984.

$$65x-x^{2}=984$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+65x=984$$

Divide both sides by -1.

$$\frac{-x^{2}+65x}{-1}=\frac{984}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{65}{-1}x=\frac{984}{-1}$$

Divide 65 by -1.

$$x^{2}-65x=\frac{984}{-1}$$

Divide 984 by -1.

$$x^{2}-65x=-984$$

Divide -65, the coefficient of the x term, by 2 to get -\frac{65}{2}. Then add the square of -\frac{65}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-65x+\left(-\frac{65}{2}\right)^{2}=-984+\left(-\frac{65}{2}\right)^{2}$$

Square -\frac{65}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-65x+\frac{4225}{4}=-984+\frac{4225}{4}$$

Add -984 to \frac{4225}{4}.

$$x^{2}-65x+\frac{4225}{4}=\frac{289}{4}$$

Factor x^{2}-65x+\frac{4225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{65}{2}\right)^{2}=\frac{289}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{65}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Simplify.

$$x-\frac{65}{2}=\frac{17}{2} x-\frac{65}{2}=-\frac{17}{2}$$

Add \frac{65}{2} to both sides of the equation.

$$x=41 x=24$$