Solve 2(x+3)*(-2)+2(x+3)^2=176

$$2 ( x + 3 ) \cdot ( - 2 ) + 2 ( x + 3 ) ^ { 2 } = 176$$

Solve for x

$$x=\sqrt{89}-2\approx 7.433981132$$

$$x=-\left(\sqrt{89}+2\right)\approx -11.433981132$$

Steps Using the Quadratic Formula

Multiply 2 and -2 to get -4.

$$-4\left(x+3\right)+2\left(x+3\right)^{2}=176$$

Use the distributive property to multiply -4 by x+3.

$$-4x-12+2\left(x+3\right)^{2}=176$$

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

$$-4x-12+2\left(x^{2}+6x+9\right)=176$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$-4x-12+2x^{2}+12x+18=176$$

Combine -4x and 12x to get 8x.

$$8x-12+2x^{2}+18=176$$

Add -12 and 18 to get 6.

$$8x+6+2x^{2}=176$$

Subtract 176 from both sides.

$$8x+6+2x^{2}-176=0$$

Subtract 176 from 6 to get -170.

$$8x-170+2x^{2}=0$$

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.

$$2x^{2}+8x-170=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -170 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\left(-170\right)}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\left(-170\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\left(-170\right)}}{2\times 2}$$

Multiply -8 times -170.

$$x=\frac{-8±\sqrt{64+1360}}{2\times 2}$$

Add 64 to 1360.

$$x=\frac{-8±\sqrt{1424}}{2\times 2}$$

Take the square root of 1424.

$$x=\frac{-8±4\sqrt{89}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±4\sqrt{89}}{4}$$

Now solve the equation x=\frac{-8±4\sqrt{89}}{4} when ± is plus. Add -8 to 4\sqrt{89}.

$$x=\frac{4\sqrt{89}-8}{4}$$

Divide -8+4\sqrt{89} by 4.

$$x=\sqrt{89}-2$$

Now solve the equation x=\frac{-8±4\sqrt{89}}{4} when ± is minus. Subtract 4\sqrt{89} from -8.

$$x=\frac{-4\sqrt{89}-8}{4}$$

Divide -8-4\sqrt{89} by 4.

$$x=-\sqrt{89}-2$$

The equation is now solved.

$$x=\sqrt{89}-2 x=-\sqrt{89}-2$$

Steps for Completing the Square

Multiply 2 and -2 to get -4.

$$-4\left(x+3\right)+2\left(x+3\right)^{2}=176$$

Use the distributive property to multiply -4 by x+3.

$$-4x-12+2\left(x+3\right)^{2}=176$$

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

$$-4x-12+2\left(x^{2}+6x+9\right)=176$$

Use the distributive property to multiply 2 by x^{2}+6x+9.

$$-4x-12+2x^{2}+12x+18=176$$

Combine -4x and 12x to get 8x.

$$8x-12+2x^{2}+18=176$$

Add -12 and 18 to get 6.

$$8x+6+2x^{2}=176$$

Subtract 6 from both sides.

$$8x+2x^{2}=176-6$$

Subtract 6 from 176 to get 170.

$$8x+2x^{2}=170$$

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.

$$2x^{2}+8x=170$$

Divide both sides by 2.

$$\frac{2x^{2}+8x}{2}=\frac{170}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{8}{2}x=\frac{170}{2}$$

Divide 8 by 2.

$$x^{2}+4x=\frac{170}{2}$$

Divide 170 by 2.

$$x^{2}+4x=85$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+4x+2^{2}=85+2^{2}$$

Square 2.

$$x^{2}+4x+4=85+4$$

Add 85 to 4.

$$x^{2}+4x+4=89$$

Factor x^{2}+4x+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}.

$$\left(x+2\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{89}$$

Simplify.

$$x+2=\sqrt{89} x+2=-\sqrt{89}$$

Subtract 2 from both sides of the equation.

$$x=\sqrt{89}-2 x=-\sqrt{89}-2$$