Solve (x-3)^2/16-(x-2)^2/4=1-x/2

$$\frac { ( x - 3 ) ^ { 2 } } { 16 } - \frac { ( x - 2 ) ^ { 2 } } { 4 } = \frac { 1 - x } { 2 }$$

Solve for x

$$x=1$$

$$x=5$$

Steps Using Factoring By Grouping

Multiply both sides of the equation by 16, the least common multiple of 16,4,2.

$$\left(x-3\right)^{2}-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x^{2}-4x+4\right)=8\left(1-x\right)$$

Use the distributive property to multiply -4 by x^{2}-4x+4.

$$x^{2}-6x+9-4x^{2}+16x-16=8\left(1-x\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+9+16x-16=8\left(1-x\right)$$

Combine -6x and 16x to get 10x.

$$-3x^{2}+10x+9-16=8\left(1-x\right)$$

Subtract 16 from 9 to get -7.

$$-3x^{2}+10x-7=8\left(1-x\right)$$

Use the distributive property to multiply 8 by 1-x.

$$-3x^{2}+10x-7=8-8x$$

Subtract 8 from both sides.

$$-3x^{2}+10x-7-8=-8x$$

Subtract 8 from -7 to get -15.

$$-3x^{2}+10x-15=-8x$$

Add 8x to both sides.

$$-3x^{2}+10x-15+8x=0$$

Combine 10x and 8x to get 18x.

$$-3x^{2}+18x-15=0$$

Divide both sides by 3.

$$-x^{2}+6x-5=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=6 ab=-\left(-5\right)=5$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=5 b=1$$

Rewrite -x^{2}+6x-5 as \left(-x^{2}+5x\right)+\left(x-5\right).

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

Factor out -x in -x^{2}+5x.

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

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x+1\right)$$

To find equation solutions, solve x-5=0 and -x+1=0.

$$x=5 x=1$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 16, the least common multiple of 16,4,2.

$$\left(x-3\right)^{2}-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x^{2}-4x+4\right)=8\left(1-x\right)$$

Use the distributive property to multiply -4 by x^{2}-4x+4.

$$x^{2}-6x+9-4x^{2}+16x-16=8\left(1-x\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+9+16x-16=8\left(1-x\right)$$

Combine -6x and 16x to get 10x.

$$-3x^{2}+10x+9-16=8\left(1-x\right)$$

Subtract 16 from 9 to get -7.

$$-3x^{2}+10x-7=8\left(1-x\right)$$

Use the distributive property to multiply 8 by 1-x.

$$-3x^{2}+10x-7=8-8x$$

Subtract 8 from both sides.

$$-3x^{2}+10x-7-8=-8x$$

Subtract 8 from -7 to get -15.

$$-3x^{2}+10x-15=-8x$$

Add 8x to both sides.

$$-3x^{2}+10x-15+8x=0$$

Combine 10x and 8x to get 18x.

$$-3x^{2}+18x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 18 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-3\right)\left(-15\right)}}{2\left(-3\right)}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\left(-3\right)\left(-15\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-18±\sqrt{324+12\left(-15\right)}}{2\left(-3\right)}$$

Multiply 12 times -15.

$$x=\frac{-18±\sqrt{324-180}}{2\left(-3\right)}$$

Add 324 to -180.

$$x=\frac{-18±\sqrt{144}}{2\left(-3\right)}$$

Take the square root of 144.

$$x=\frac{-18±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-18±12}{-6}$$

Now solve the equation x=\frac{-18±12}{-6} when ± is plus. Add -18 to 12.

$$x=-\frac{6}{-6}$$

Divide -6 by -6.

$$x=1$$

Now solve the equation x=\frac{-18±12}{-6} when ± is minus. Subtract 12 from -18.

$$x=-\frac{30}{-6}$$

Divide -30 by -6.

$$x=5$$

The equation is now solved.

$$x=1 x=5$$

Steps for Completing the Square

Multiply both sides of the equation by 16, the least common multiple of 16,4,2.

$$\left(x-3\right)^{2}-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x-2\right)^{2}=8\left(1-x\right)$$

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

$$x^{2}-6x+9-4\left(x^{2}-4x+4\right)=8\left(1-x\right)$$

Use the distributive property to multiply -4 by x^{2}-4x+4.

$$x^{2}-6x+9-4x^{2}+16x-16=8\left(1-x\right)$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-6x+9+16x-16=8\left(1-x\right)$$

Combine -6x and 16x to get 10x.

$$-3x^{2}+10x+9-16=8\left(1-x\right)$$

Subtract 16 from 9 to get -7.

$$-3x^{2}+10x-7=8\left(1-x\right)$$

Use the distributive property to multiply 8 by 1-x.

$$-3x^{2}+10x-7=8-8x$$

Add 8x to both sides.

$$-3x^{2}+10x-7+8x=8$$

Combine 10x and 8x to get 18x.

$$-3x^{2}+18x-7=8$$

Add 7 to both sides.

$$-3x^{2}+18x=8+7$$

Add 8 and 7 to get 15.

$$-3x^{2}+18x=15$$

Divide both sides by -3.

$$\frac{-3x^{2}+18x}{-3}=\frac{15}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{18}{-3}x=\frac{15}{-3}$$

Divide 18 by -3.

$$x^{2}-6x=\frac{15}{-3}$$

Divide 15 by -3.

$$x^{2}-6x=-5$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-5+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-5+9$$

Add -5 to 9.

$$x^{2}-6x+9=4$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-3=2 x-3=-2$$

Add 3 to both sides of the equation.

$$x=5 x=1$$