Solve v^2-7v/6-3=1

$$\frac { v ^ { 2 } - 7 v } { 6 } - 3 = 1$$

Solve for v

$$v = \frac{\sqrt{145} + 7}{2} \approx 9.520797289$$

$$v=\frac{7-\sqrt{145}}{2}\approx -2.520797289$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6.

$$v^{2}-7v-18=6$$

Subtract 6 from both sides.

$$v^{2}-7v-18-6=0$$

Subtract 6 from -18 to get -24.

$$v^{2}-7v-24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -7 for b, and -24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-24\right)}}{2}$$

Square -7.

$$v=\frac{-\left(-7\right)±\sqrt{49-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$v=\frac{-\left(-7\right)±\sqrt{49+96}}{2}$$

Add 49 to 96.

$$v=\frac{-\left(-7\right)±\sqrt{145}}{2}$$

The opposite of -7 is 7.

$$v=\frac{7±\sqrt{145}}{2}$$

Now solve the equation v=\frac{7±\sqrt{145}}{2} when ± is plus. Add 7 to \sqrt{145}.

$$v=\frac{\sqrt{145}+7}{2}$$

Now solve the equation v=\frac{7±\sqrt{145}}{2} when ± is minus. Subtract \sqrt{145} from 7.

$$v=\frac{7-\sqrt{145}}{2}$$

The equation is now solved.

$$v=\frac{\sqrt{145}+7}{2} v=\frac{7-\sqrt{145}}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 6.

$$v^{2}-7v-18=6$$

Add 18 to both sides.

$$v^{2}-7v=6+18$$

Add 6 and 18 to get 24.

$$v^{2}-7v=24$$

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.

$$v^{2}-7v+\left(-\frac{7}{2}\right)^{2}=24+\left(-\frac{7}{2}\right)^{2}$$

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

$$v^{2}-7v+\frac{49}{4}=24+\frac{49}{4}$$

Add 24 to \frac{49}{4}.

$$v^{2}-7v+\frac{49}{4}=\frac{145}{4}$$

Factor v^{2}-7v+\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}.

$$\left(v-\frac{7}{2}\right)^{2}=\frac{145}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{7}{2}\right)^{2}}=\sqrt{\frac{145}{4}}$$

Simplify.

$$v-\frac{7}{2}=\frac{\sqrt{145}}{2} v-\frac{7}{2}=-\frac{\sqrt{145}}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$v=\frac{\sqrt{145}+7}{2} v=\frac{7-\sqrt{145}}{2}$$