Solve left(16-2xright)left(12-2xright)=16*121/2

$$\left( 16-2x \right) \left( 12-2x \right) = 16 \times 12 \frac{ 1 }{ 2 }$$

Solve for x

$$x=7-\sqrt{51}\approx -0.141428429$$

$$x=\sqrt{51}+7\approx 14.141428429$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$2\left(16-2x\right)\left(12-2x\right)=16\left(12\times 2+1\right)$$

Use the distributive property to multiply 2 by 16-2x.

$$\left(32-4x\right)\left(12-2x\right)=16\left(12\times 2+1\right)$$

Use the distributive property to multiply 32-4x by 12-2x and combine like terms.

$$384-112x+8x^{2}=16\left(12\times 2+1\right)$$

Multiply 12 and 2 to get 24.

$$384-112x+8x^{2}=16\left(24+1\right)$$

Add 24 and 1 to get 25.

$$384-112x+8x^{2}=16\times 25$$

Multiply 16 and 25 to get 400.

$$384-112x+8x^{2}=400$$

Subtract 400 from both sides.

$$384-112x+8x^{2}-400=0$$

Subtract 400 from 384 to get -16.

$$-16-112x+8x^{2}=0$$

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.

$$8x^{2}-112x-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -112 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-112\right)±\sqrt{\left(-112\right)^{2}-4\times 8\left(-16\right)}}{2\times 8}$$

Square -112.

$$x=\frac{-\left(-112\right)±\sqrt{12544-4\times 8\left(-16\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-112\right)±\sqrt{12544-32\left(-16\right)}}{2\times 8}$$

Multiply -32 times -16.

$$x=\frac{-\left(-112\right)±\sqrt{12544+512}}{2\times 8}$$

Add 12544 to 512.

$$x=\frac{-\left(-112\right)±\sqrt{13056}}{2\times 8}$$

Take the square root of 13056.

$$x=\frac{-\left(-112\right)±16\sqrt{51}}{2\times 8}$$

The opposite of -112 is 112.

$$x=\frac{112±16\sqrt{51}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{112±16\sqrt{51}}{16}$$

Now solve the equation x=\frac{112±16\sqrt{51}}{16} when ± is plus. Add 112 to 16\sqrt{51}.

$$x=\frac{16\sqrt{51}+112}{16}$$

Divide 112+16\sqrt{51} by 16.

$$x=\sqrt{51}+7$$

Now solve the equation x=\frac{112±16\sqrt{51}}{16} when ± is minus. Subtract 16\sqrt{51} from 112.

$$x=\frac{112-16\sqrt{51}}{16}$$

Divide 112-16\sqrt{51} by 16.

$$x=7-\sqrt{51}$$

The equation is now solved.

$$x=\sqrt{51}+7 x=7-\sqrt{51}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$2\left(16-2x\right)\left(12-2x\right)=16\left(12\times 2+1\right)$$

Use the distributive property to multiply 2 by 16-2x.

$$\left(32-4x\right)\left(12-2x\right)=16\left(12\times 2+1\right)$$

Use the distributive property to multiply 32-4x by 12-2x and combine like terms.

$$384-112x+8x^{2}=16\left(12\times 2+1\right)$$

Multiply 12 and 2 to get 24.

$$384-112x+8x^{2}=16\left(24+1\right)$$

Add 24 and 1 to get 25.

$$384-112x+8x^{2}=16\times 25$$

Multiply 16 and 25 to get 400.

$$384-112x+8x^{2}=400$$

Subtract 384 from both sides.

$$-112x+8x^{2}=400-384$$

Subtract 384 from 400 to get 16.

$$-112x+8x^{2}=16$$

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.

$$8x^{2}-112x=16$$

Divide both sides by 8.

$$\frac{8x^{2}-112x}{8}=\frac{16}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{112}{8}\right)x=\frac{16}{8}$$

Divide -112 by 8.

$$x^{2}-14x=\frac{16}{8}$$

Divide 16 by 8.

$$x^{2}-14x=2$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=2+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=2+49$$

Add 2 to 49.

$$x^{2}-14x+49=51$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=51$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{51}$$

Simplify.

$$x-7=\sqrt{51} x-7=-\sqrt{51}$$

Add 7 to both sides of the equation.

$$x=\sqrt{51}+7 x=7-\sqrt{51}$$