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

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

Solve for x

$$x=-1$$

$$x = \frac{7}{2} = 3\frac{1}{2} = 3.5$$

Steps Using the Quadratic Formula

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

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

Subtract 4 from both sides.

$$2x^{2}-5x-3-4=0$$

Subtract 4 from -3 to get -7.

$$2x^{2}-5x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -5 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-5\right)±\sqrt{25-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-5\right)±\sqrt{25+56}}{2\times 2}$$

Add 25 to 56.

$$x=\frac{-\left(-5\right)±\sqrt{81}}{2\times 2}$$

Take the square root of 81.

$$x=\frac{-\left(-5\right)±9}{2\times 2}$$

The opposite of -5 is 5.

$$x=\frac{5±9}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{5±9}{4}$$

Now solve the equation x=\frac{5±9}{4} when ± is plus. Add 5 to 9.

$$x=\frac{14}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

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

Now solve the equation x=\frac{5±9}{4} when ± is minus. Subtract 9 from 5.

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

Divide -4 by 4.

$$x=-1$$

The equation is now solved.

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

Steps for Completing the Square

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

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

Add 3 to both sides.

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

Add 4 and 3 to get 7.

$$2x^{2}-5x=7$$

Divide both sides by 2.

$$\frac{2x^{2}-5x}{2}=\frac{7}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{5}{2}x=\frac{7}{2}$$

Divide -\frac{5}{2}, the coefficient of the x term, by 2 to get -\frac{5}{4}. Then add the square of -\frac{5}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{5}{2}x+\left(-\frac{5}{4}\right)^{2}=\frac{7}{2}+\left(-\frac{5}{4}\right)^{2}$$

Square -\frac{5}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{7}{2}+\frac{25}{16}$$

Add \frac{7}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{5}{2}x+\frac{25}{16}=\frac{81}{16}$$

Factor x^{2}-\frac{5}{2}x+\frac{25}{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}.

$$\left(x-\frac{5}{4}\right)^{2}=\frac{81}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Simplify.

$$x-\frac{5}{4}=\frac{9}{4} x-\frac{5}{4}=-\frac{9}{4}$$

Add \frac{5}{4} to both sides of the equation.

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