Solve (x+1)(x-3)/2+x=x/4

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

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=2$$

Steps Using Factoring By Grouping

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

$$2\left(x+1\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2 by x+1.

$$\left(2x+2\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2x+2 by x-3 and combine like terms.

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

Combine -4x and 4x to get 0.

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

Subtract x from both sides.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 2x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=-1 ab=2\left(-6\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -1.

$$a=-4 b=3$$

Rewrite 2x^{2}-x-6 as \left(2x^{2}-4x\right)+\left(3x-6\right).

$$\left(2x^{2}-4x\right)+\left(3x-6\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-2\right)+3\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(2x+3\right)$$

To find equation solutions, solve x-2=0 and 2x+3=0.

$$x=2 x=-\frac{3}{2}$$

Steps Using the Quadratic Formula

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

$$2\left(x+1\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2 by x+1.

$$\left(2x+2\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2x+2 by x-3 and combine like terms.

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

Combine -4x and 4x to get 0.

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

Subtract x from both sides.

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

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.

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

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

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-6\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-6\right)}}{2\times 2}$$

Multiply -8 times -6.

$$x=\frac{-\left(-1\right)±\sqrt{1+48}}{2\times 2}$$

Add 1 to 48.

$$x=\frac{-\left(-1\right)±\sqrt{49}}{2\times 2}$$

Take the square root of 49.

$$x=\frac{-\left(-1\right)±7}{2\times 2}$$

The opposite of -1 is 1.

$$x=\frac{1±7}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{1±7}{4}$$

Now solve the equation x=\frac{1±7}{4} when ± is plus. Add 1 to 7.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{1±7}{4} when ± is minus. Subtract 7 from 1.

$$x=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

The equation is now solved.

$$x=2 x=-\frac{3}{2}$$

Steps for Completing the Square

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

$$2\left(x+1\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2 by x+1.

$$\left(2x+2\right)\left(x-3\right)+4x=x$$

Use the distributive property to multiply 2x+2 by x-3 and combine like terms.

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

Combine -4x and 4x to get 0.

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

Subtract x from both sides.

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

Add 6 to both sides. Anything plus zero gives itself.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{1}{2}x=\frac{6}{2}$$

Divide 6 by 2.

$$x^{2}-\frac{1}{2}x=3$$

Divide -\frac{1}{2}, the coefficient of the x term, by 2 to get -\frac{1}{4}. Then add the square of -\frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -\frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=3+\frac{1}{16}$$

Add 3 to \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{49}{16}$$

Factor x^{2}-\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{49}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{49}{16}}$$

Simplify.

$$x-\frac{1}{4}=\frac{7}{4} x-\frac{1}{4}=-\frac{7}{4}$$

Add \frac{1}{4} to both sides of the equation.

$$x=2 x=-\frac{3}{2}$$