Solve 5a^2-7a=2a(a-3)

$$5 a ^ { 2 } - 7 a = 2 a ( a - 3 )$$

Solve for a

$$a=\frac{1}{3}\approx 0.333333333$$

$$a=0$$

Steps Using Factoring

Use the distributive property to multiply 2a by a-3.

$$5a^{2}-7a=2a^{2}-6a$$

Subtract 2a^{2} from both sides.

$$5a^{2}-7a-2a^{2}=-6a$$

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

$$3a^{2}-7a=-6a$$

Add 6a to both sides.

$$3a^{2}-7a+6a=0$$

Combine -7a and 6a to get -a.

$$3a^{2}-a=0$$

Factor out a.

$$a\left(3a-1\right)=0$$

To find equation solutions, solve a=0 and 3a-1=0.

$$a=0 a=\frac{1}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 2a by a-3.

$$5a^{2}-7a=2a^{2}-6a$$

Subtract 2a^{2} from both sides.

$$5a^{2}-7a-2a^{2}=-6a$$

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

$$3a^{2}-7a=-6a$$

Add 6a to both sides.

$$3a^{2}-7a+6a=0$$

Combine -7a and 6a to get -a.

$$3a^{2}-a=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-1\right)±\sqrt{1}}{2\times 3}$$

Take the square root of 1.

$$a=\frac{-\left(-1\right)±1}{2\times 3}$$

The opposite of -1 is 1.

$$a=\frac{1±1}{2\times 3}$$

Multiply 2 times 3.

$$a=\frac{1±1}{6}$$

Now solve the equation a=\frac{1±1}{6} when ± is plus. Add 1 to 1.

$$a=\frac{2}{6}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$a=\frac{1}{3}$$

Now solve the equation a=\frac{1±1}{6} when ± is minus. Subtract 1 from 1.

$$a=\frac{0}{6}$$

Divide 0 by 6.

$$a=0$$

The equation is now solved.

$$a=\frac{1}{3} a=0$$

Steps for Completing the Square

Use the distributive property to multiply 2a by a-3.

$$5a^{2}-7a=2a^{2}-6a$$

Subtract 2a^{2} from both sides.

$$5a^{2}-7a-2a^{2}=-6a$$

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

$$3a^{2}-7a=-6a$$

Add 6a to both sides.

$$3a^{2}-7a+6a=0$$

Combine -7a and 6a to get -a.

$$3a^{2}-a=0$$

Divide both sides by 3.

$$\frac{3a^{2}-a}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$a^{2}-\frac{1}{3}a=\frac{0}{3}$$

Divide 0 by 3.

$$a^{2}-\frac{1}{3}a=0$$

Divide -\frac{1}{3}, the coefficient of the x term, by 2 to get -\frac{1}{6}. Then add the square of -\frac{1}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-\frac{1}{3}a+\left(-\frac{1}{6}\right)^{2}=\left(-\frac{1}{6}\right)^{2}$$

Square -\frac{1}{6} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-\frac{1}{3}a+\frac{1}{36}=\frac{1}{36}$$

Factor a^{2}-\frac{1}{3}a+\frac{1}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-\frac{1}{6}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{1}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$a-\frac{1}{6}=\frac{1}{6} a-\frac{1}{6}=-\frac{1}{6}$$

Add \frac{1}{6} to both sides of the equation.

$$a=\frac{1}{3} a=0$$