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

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

Solve for x

$$x=\frac{3\sqrt{2}}{2}+1\approx 3.121320344$$

$$x=-\frac{3\sqrt{2}}{2}+1\approx -1.121320344$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2x+1 by 3-x and combine like terms.

$$x-\left(5x-2x^{2}+3\right)=4$$

To find the opposite of 5x-2x^{2}+3, find the opposite of each term.

$$x-5x+2x^{2}-3=4$$

Combine x and -5x to get -4x.

$$-4x+2x^{2}-3=4$$

Subtract 4 from both sides.

$$-4x+2x^{2}-3-4=0$$

Subtract 4 from -3 to get -7.

$$-4x+2x^{2}-7=0$$

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.

$$2x^{2}-4x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-4\right)±\sqrt{16+56}}{2\times 2}$$

Add 16 to 56.

$$x=\frac{-\left(-4\right)±\sqrt{72}}{2\times 2}$$

Take the square root of 72.

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

The opposite of -4 is 4.

$$x=\frac{4±6\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±6\sqrt{2}}{4}$$

Now solve the equation x=\frac{4±6\sqrt{2}}{4} when ± is plus. Add 4 to 6\sqrt{2}.

$$x=\frac{6\sqrt{2}+4}{4}$$

Divide 4+6\sqrt{2} by 4.

$$x=\frac{3\sqrt{2}}{2}+1$$

Now solve the equation x=\frac{4±6\sqrt{2}}{4} when ± is minus. Subtract 6\sqrt{2} from 4.

$$x=\frac{4-6\sqrt{2}}{4}$$

Divide 4-6\sqrt{2} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use the distributive property to multiply 2x+1 by 3-x and combine like terms.

$$x-\left(5x-2x^{2}+3\right)=4$$

To find the opposite of 5x-2x^{2}+3, find the opposite of each term.

$$x-5x+2x^{2}-3=4$$

Combine x and -5x to get -4x.

$$-4x+2x^{2}-3=4$$

Add 3 to both sides.

$$-4x+2x^{2}=4+3$$

Add 4 and 3 to get 7.

$$-4x+2x^{2}=7$$

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.

$$2x^{2}-4x=7$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{7}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{7}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{7}{2}$$

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

$$x^{2}-2x+1=\frac{7}{2}+1$$

Add \frac{7}{2} to 1.

$$x^{2}-2x+1=\frac{9}{2}$$

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

$$\left(x-1\right)^{2}=\frac{9}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{9}{2}}$$

Simplify.

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

Add 1 to both sides of the equation.

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