Solve 2v(v-7)=5v(v-7)

$$2 v ( v - 7 ) = 5 v ( v - 7 )$$

Solve for v

$$v=7$$

$$v=0$$

Steps Using Factoring

Use the distributive property to multiply 2v by v-7.

$$2v^{2}-14v=5v\left(v-7\right)$$

Use the distributive property to multiply 5v by v-7.

$$2v^{2}-14v=5v^{2}-35v$$

Subtract 5v^{2} from both sides.

$$2v^{2}-14v-5v^{2}=-35v$$

Combine 2v^{2} and -5v^{2} to get -3v^{2}.

$$-3v^{2}-14v=-35v$$

Add 35v to both sides.

$$-3v^{2}-14v+35v=0$$

Combine -14v and 35v to get 21v.

$$-3v^{2}+21v=0$$

Factor out v.

$$v\left(-3v+21\right)=0$$

To find equation solutions, solve v=0 and -3v+21=0.

$$v=0 v=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2v by v-7.

$$2v^{2}-14v=5v\left(v-7\right)$$

Use the distributive property to multiply 5v by v-7.

$$2v^{2}-14v=5v^{2}-35v$$

Subtract 5v^{2} from both sides.

$$2v^{2}-14v-5v^{2}=-35v$$

Combine 2v^{2} and -5v^{2} to get -3v^{2}.

$$-3v^{2}-14v=-35v$$

Add 35v to both sides.

$$-3v^{2}-14v+35v=0$$

Combine -14v and 35v to get 21v.

$$-3v^{2}+21v=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-21±\sqrt{21^{2}}}{2\left(-3\right)}$$

Take the square root of 21^{2}.

$$v=\frac{-21±21}{2\left(-3\right)}$$

Multiply 2 times -3.

$$v=\frac{-21±21}{-6}$$

Now solve the equation v=\frac{-21±21}{-6} when ± is plus. Add -21 to 21.

$$v=\frac{0}{-6}$$

Divide 0 by -6.

$$v=0$$

Now solve the equation v=\frac{-21±21}{-6} when ± is minus. Subtract 21 from -21.

$$v=-\frac{42}{-6}$$

Divide -42 by -6.

$$v=7$$

The equation is now solved.

$$v=0 v=7$$

Steps for Completing the Square

Use the distributive property to multiply 2v by v-7.

$$2v^{2}-14v=5v\left(v-7\right)$$

Use the distributive property to multiply 5v by v-7.

$$2v^{2}-14v=5v^{2}-35v$$

Subtract 5v^{2} from both sides.

$$2v^{2}-14v-5v^{2}=-35v$$

Combine 2v^{2} and -5v^{2} to get -3v^{2}.

$$-3v^{2}-14v=-35v$$

Add 35v to both sides.

$$-3v^{2}-14v+35v=0$$

Combine -14v and 35v to get 21v.

$$-3v^{2}+21v=0$$

Divide both sides by -3.

$$\frac{-3v^{2}+21v}{-3}=\frac{0}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$v^{2}+\frac{21}{-3}v=\frac{0}{-3}$$

Divide 21 by -3.

$$v^{2}-7v=\frac{0}{-3}$$

Divide 0 by -3.

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

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.

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

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

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

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

$$v=7 v=0$$