Solve 2(x-3)^2=200

$$2 ( x - 3 ) ^ { 2 } = 200$$

Solve for x

$$x=13$$

$$x=-7$$

Steps Using Factoring

Divide both sides by 2.

$$\left(x-3\right)^{2}=\frac{200}{2}$$

Divide 200 by 2 to get 100.

$$\left(x-3\right)^{2}=100$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=100$$

Subtract 100 from both sides.

$$x^{2}-6x+9-100=0$$

Subtract 100 from 9 to get -91.

$$x^{2}-6x-91=0$$

To solve the equation, factor x^{2}-6x-91 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=-6 ab=-91$$

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 -91.

$$1,-91 7,-13$$

Calculate the sum for each pair.

$$1-91=-90 7-13=-6$$

The solution is the pair that gives sum -6.

$$a=-13 b=7$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x-13\right)\left(x+7\right)$$

To find equation solutions, solve x-13=0 and x+7=0.

$$x=13 x=-7$$

Steps Using Factoring By Grouping

Divide both sides by 2.

$$\left(x-3\right)^{2}=\frac{200}{2}$$

Divide 200 by 2 to get 100.

$$\left(x-3\right)^{2}=100$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=100$$

Subtract 100 from both sides.

$$x^{2}-6x+9-100=0$$

Subtract 100 from 9 to get -91.

$$x^{2}-6x-91=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-91. To find a and b, set up a system to be solved.

$$a+b=-6 ab=1\left(-91\right)=-91$$

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 -91.

$$1,-91 7,-13$$

Calculate the sum for each pair.

$$1-91=-90 7-13=-6$$

The solution is the pair that gives sum -6.

$$a=-13 b=7$$

Rewrite x^{2}-6x-91 as \left(x^{2}-13x\right)+\left(7x-91\right).

$$\left(x^{2}-13x\right)+\left(7x-91\right)$$

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

$$x\left(x-13\right)+7\left(x-13\right)$$

Factor out common term x-13 by using distributive property.

$$\left(x-13\right)\left(x+7\right)$$

To find equation solutions, solve x-13=0 and x+7=0.

$$x=13 x=-7$$

Steps Using the Quadratic Formula

Divide both sides by 2.

$$\left(x-3\right)^{2}=\frac{200}{2}$$

Divide 200 by 2 to get 100.

$$\left(x-3\right)^{2}=100$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

$$x^{2}-6x+9=100$$

Subtract 100 from both sides.

$$x^{2}-6x+9-100=0$$

Subtract 100 from 9 to get -91.

$$x^{2}-6x-91=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -6 for b, and -91 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-91\right)}}{2}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-91\right)}}{2}$$

Multiply -4 times -91.

$$x=\frac{-\left(-6\right)±\sqrt{36+364}}{2}$$

Add 36 to 364.

$$x=\frac{-\left(-6\right)±\sqrt{400}}{2}$$

Take the square root of 400.

$$x=\frac{-\left(-6\right)±20}{2}$$

The opposite of -6 is 6.

$$x=\frac{6±20}{2}$$

Now solve the equation x=\frac{6±20}{2} when ± is plus. Add 6 to 20.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{6±20}{2} when ± is minus. Subtract 20 from 6.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=13 x=-7$$

Steps for Completing the Square

Divide both sides by 2.

$$\left(x-3\right)^{2}=\frac{200}{2}$$

Divide 200 by 2 to get 100.

$$\left(x-3\right)^{2}=100$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{100}$$

Simplify.

$$x-3=10 x-3=-10$$

Add 3 to both sides of the equation.

$$x=13 x=-7$$