Solve x(x+7)/2=34

$$\frac { x ( x + 7 ) } { 2 } = 34$$

Solve for x

$$x = \frac{\sqrt{321} - 7}{2} \approx 5.458236434$$

$$x=\frac{-\sqrt{321}-7}{2}\approx -12.458236434$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$x\left(x+7\right)=34\times 2$$

Use the distributive property to multiply x by x+7.

$$x^{2}+7x=34\times 2$$

Multiply 34 and 2 to get 68.

$$x^{2}+7x=68$$

Subtract 68 from both sides.

$$x^{2}+7x-68=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 7 for b, and -68 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-68\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-68\right)}}{2}$$

Multiply -4 times -68.

$$x=\frac{-7±\sqrt{49+272}}{2}$$

Add 49 to 272.

$$x=\frac{-7±\sqrt{321}}{2}$$

Now solve the equation x=\frac{-7±\sqrt{321}}{2} when ± is plus. Add -7 to \sqrt{321}.

$$x=\frac{\sqrt{321}-7}{2}$$

Now solve the equation x=\frac{-7±\sqrt{321}}{2} when ± is minus. Subtract \sqrt{321} from -7.

$$x=\frac{-\sqrt{321}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{321}-7}{2} x=\frac{-\sqrt{321}-7}{2}$$

Steps for Completing the Square

Multiply both sides by 2.

$$x\left(x+7\right)=34\times 2$$

Use the distributive property to multiply x by x+7.

$$x^{2}+7x=34\times 2$$

Multiply 34 and 2 to get 68.

$$x^{2}+7x=68$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=68+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=68+\frac{49}{4}$$

Add 68 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{321}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{321}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{321}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{321}}{2} x+\frac{7}{2}=-\frac{\sqrt{321}}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{\sqrt{321}-7}{2} x=\frac{-\sqrt{321}-7}{2}$$