Solve x(32+(x-1)(-4))/2=x(23+(x-1)(-1))/2

$$\frac{ x(32+(x-1)(-4)) }{ 2 } = \frac{ x(23+(x-1)(-1)) }{ 2 }$$

Solve for x

$$x=4$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 2.

$$x\left(32+\left(x-1\right)\left(-4\right)\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x-1 by -4.

$$x\left(32-4x+4\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Add 32 and 4 to get 36.

$$x\left(36-4x\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x by 36-4x.

$$36x-4x^{2}=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

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

$$36x-4x^{2}=x\left(23-x+1\right)$$

Add 23 and 1 to get 24.

$$36x-4x^{2}=x\left(24-x\right)$$

Use the distributive property to multiply x by 24-x.

$$36x-4x^{2}=24x-x^{2}$$

Subtract 24x from both sides.

$$36x-4x^{2}-24x=-x^{2}$$

Combine 36x and -24x to get 12x.

$$12x-4x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$12x-4x^{2}+x^{2}=0$$

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

$$12x-3x^{2}=0$$

Factor out x.

$$x\left(12-3x\right)=0$$

To find equation solutions, solve x=0 and 12-3x=0.

$$x=0 x=4$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(32+\left(x-1\right)\left(-4\right)\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x-1 by -4.

$$x\left(32-4x+4\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Add 32 and 4 to get 36.

$$x\left(36-4x\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x by 36-4x.

$$36x-4x^{2}=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

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

$$36x-4x^{2}=x\left(23-x+1\right)$$

Add 23 and 1 to get 24.

$$36x-4x^{2}=x\left(24-x\right)$$

Use the distributive property to multiply x by 24-x.

$$36x-4x^{2}=24x-x^{2}$$

Subtract 24x from both sides.

$$36x-4x^{2}-24x=-x^{2}$$

Combine 36x and -24x to get 12x.

$$12x-4x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$12x-4x^{2}+x^{2}=0$$

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

$$12x-3x^{2}=0$$

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.

$$-3x^{2}+12x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 12 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}}}{2\left(-3\right)}$$

Take the square root of 12^{2}.

$$x=\frac{-12±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-12±12}{-6}$$

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

$$x=\frac{0}{-6}$$

Divide 0 by -6.

$$x=0$$

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

$$x=-\frac{24}{-6}$$

Divide -24 by -6.

$$x=4$$

The equation is now solved.

$$x=0 x=4$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(32+\left(x-1\right)\left(-4\right)\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x-1 by -4.

$$x\left(32-4x+4\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Add 32 and 4 to get 36.

$$x\left(36-4x\right)=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

Use the distributive property to multiply x by 36-4x.

$$36x-4x^{2}=x\left(23+\left(x-1\right)\left(-1\right)\right)$$

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

$$36x-4x^{2}=x\left(23-x+1\right)$$

Add 23 and 1 to get 24.

$$36x-4x^{2}=x\left(24-x\right)$$

Use the distributive property to multiply x by 24-x.

$$36x-4x^{2}=24x-x^{2}$$

Subtract 24x from both sides.

$$36x-4x^{2}-24x=-x^{2}$$

Combine 36x and -24x to get 12x.

$$12x-4x^{2}=-x^{2}$$

Add x^{2} to both sides.

$$12x-4x^{2}+x^{2}=0$$

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

$$12x-3x^{2}=0$$

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.

$$-3x^{2}+12x=0$$

Divide both sides by -3.

$$\frac{-3x^{2}+12x}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{12}{-3}x=\frac{0}{-3}$$

Divide 12 by -3.

$$x^{2}-4x=\frac{0}{-3}$$

Divide 0 by -3.

$$x^{2}-4x=0$$

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.

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

Square -2.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Add 2 to both sides of the equation.

$$x=4 x=0$$