Solve (x+x+3)(x+1)/2=39

$$\frac { ( x + x + 3 ) ( x + 1 ) } { 2 } = 39$$

Solve for x

$$x = -\frac{15}{2} = -7\frac{1}{2} = -7.5$$

$$x=5$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$\left(x+x+3\right)\left(x+1\right)=39\times 2$$

Combine x and x to get 2x.

$$\left(2x+3\right)\left(x+1\right)=39\times 2$$

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

$$2x^{2}+5x+3=39\times 2$$

Multiply 39 and 2 to get 78.

$$2x^{2}+5x+3=78$$

Subtract 78 from both sides.

$$2x^{2}+5x+3-78=0$$

Subtract 78 from 3 to get -75.

$$2x^{2}+5x-75=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 5 for b, and -75 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 2\left(-75\right)}}{2\times 2}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 2\left(-75\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-5±\sqrt{25-8\left(-75\right)}}{2\times 2}$$

Multiply -8 times -75.

$$x=\frac{-5±\sqrt{25+600}}{2\times 2}$$

Add 25 to 600.

$$x=\frac{-5±\sqrt{625}}{2\times 2}$$

Take the square root of 625.

$$x=\frac{-5±25}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-5±25}{4}$$

Now solve the equation x=\frac{-5±25}{4} when ± is plus. Add -5 to 25.

$$x=\frac{20}{4}$$

Divide 20 by 4.

$$x=5$$

Now solve the equation x=\frac{-5±25}{4} when ± is minus. Subtract 25 from -5.

$$x=-\frac{30}{4}$$

Reduce the fraction \frac{-30}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{15}{2}$$

The equation is now solved.

$$x=5 x=-\frac{15}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$\left(x+x+3\right)\left(x+1\right)=39\times 2$$

Combine x and x to get 2x.

$$\left(2x+3\right)\left(x+1\right)=39\times 2$$

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

$$2x^{2}+5x+3=39\times 2$$

Multiply 39 and 2 to get 78.

$$2x^{2}+5x+3=78$$

Subtract 3 from both sides.

$$2x^{2}+5x=78-3$$

Subtract 3 from 78 to get 75.

$$2x^{2}+5x=75$$

Divide both sides by 2.

$$\frac{2x^{2}+5x}{2}=\frac{75}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{5}{2}x=\frac{75}{2}$$

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.

$$x^{2}+\frac{5}{2}x+\left(\frac{5}{4}\right)^{2}=\frac{75}{2}+\left(\frac{5}{4}\right)^{2}$$

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

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{75}{2}+\frac{25}{16}$$

Add \frac{75}{2} to \frac{25}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{5}{2}x+\frac{25}{16}=\frac{625}{16}$$

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}.

$$\left(x+\frac{5}{4}\right)^{2}=\frac{625}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{4}\right)^{2}}=\sqrt{\frac{625}{16}}$$

Simplify.

$$x+\frac{5}{4}=\frac{25}{4} x+\frac{5}{4}=-\frac{25}{4}$$

Subtract \frac{5}{4} from both sides of the equation.

$$x=5 x=-\frac{15}{2}$$