Solve 16^2+32x-2x^2=366

$$16 ^ { 2 } + 32 x - 2 x ^ { 2 } = 366$$

Solve for x

$$x=5$$

$$x=11$$

Steps Using Factoring By Grouping

Calculate 16 to the power of 2 and get 256.

$$256+32x-2x^{2}=366$$

Subtract 366 from both sides.

$$256+32x-2x^{2}-366=0$$

Subtract 366 from 256 to get -110.

$$-110+32x-2x^{2}=0$$

Divide both sides by 2.

$$-55+16x-x^{2}=0$$

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

$$-x^{2}+16x-55=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-55. To find a and b, set up a system to be solved.

$$a+b=16 ab=-\left(-55\right)=55$$

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 55.

$$1,55 5,11$$

Calculate the sum for each pair.

$$1+55=56 5+11=16$$

The solution is the pair that gives sum 16.

$$a=11 b=5$$

Rewrite -x^{2}+16x-55 as \left(-x^{2}+11x\right)+\left(5x-55\right).

$$\left(-x^{2}+11x\right)+\left(5x-55\right)$$

Factor out -x in the first and 5 in the second group.

$$-x\left(x-11\right)+5\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(-x+5\right)$$

To find equation solutions, solve x-11=0 and -x+5=0.

$$x=11 x=5$$

Steps Using the Quadratic Formula

Calculate 16 to the power of 2 and get 256.

$$256+32x-2x^{2}=366$$

Subtract 366 from both sides.

$$256+32x-2x^{2}-366=0$$

Subtract 366 from 256 to get -110.

$$-110+32x-2x^{2}=0$$

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.

$$-2x^{2}+32x-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2 for a, 32 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-2\right)\left(-110\right)}}{2\left(-2\right)}$$

Square 32.

$$x=\frac{-32±\sqrt{1024-4\left(-2\right)\left(-110\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$x=\frac{-32±\sqrt{1024+8\left(-110\right)}}{2\left(-2\right)}$$

Multiply 8 times -110.

$$x=\frac{-32±\sqrt{1024-880}}{2\left(-2\right)}$$

Add 1024 to -880.

$$x=\frac{-32±\sqrt{144}}{2\left(-2\right)}$$

Take the square root of 144.

$$x=\frac{-32±12}{2\left(-2\right)}$$

Multiply 2 times -2.

$$x=\frac{-32±12}{-4}$$

Now solve the equation x=\frac{-32±12}{-4} when ± is plus. Add -32 to 12.

$$x=-\frac{20}{-4}$$

Divide -20 by -4.

$$x=5$$

Now solve the equation x=\frac{-32±12}{-4} when ± is minus. Subtract 12 from -32.

$$x=-\frac{44}{-4}$$

Divide -44 by -4.

$$x=11$$

The equation is now solved.

$$x=5 x=11$$

Steps for Completing the Square

Calculate 16 to the power of 2 and get 256.

$$256+32x-2x^{2}=366$$

Subtract 256 from both sides.

$$32x-2x^{2}=366-256$$

Subtract 256 from 366 to get 110.

$$32x-2x^{2}=110$$

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.

$$-2x^{2}+32x=110$$

Divide both sides by -2.

$$\frac{-2x^{2}+32x}{-2}=\frac{110}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$x^{2}+\frac{32}{-2}x=\frac{110}{-2}$$

Divide 32 by -2.

$$x^{2}-16x=\frac{110}{-2}$$

Divide 110 by -2.

$$x^{2}-16x=-55$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-55+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-55+64$$

Add -55 to 64.

$$x^{2}-16x+64=9$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{9}$$

Simplify.

$$x-8=3 x-8=-3$$

Add 8 to both sides of the equation.

$$x=11 x=5$$