Solve 3a^2-4a=8*(2a^2-7a-3)

$$3 a ^ { 2 } - 4 a = 8 * ( 2 a ^ { 2 } - 7 a - 3 )$$

Solve for a

$$a=\frac{2\sqrt{247}}{13}+2\approx 4.417882099$$

$$a=-\frac{2\sqrt{247}}{13}+2\approx -0.417882099$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8 by 2a^{2}-7a-3.

$$3a^{2}-4a=16a^{2}-56a-24$$

Subtract 16a^{2} from both sides.

$$3a^{2}-4a-16a^{2}=-56a-24$$

Combine 3a^{2} and -16a^{2} to get -13a^{2}.

$$-13a^{2}-4a=-56a-24$$

Add 56a to both sides.

$$-13a^{2}-4a+56a=-24$$

Combine -4a and 56a to get 52a.

$$-13a^{2}+52a=-24$$

Add 24 to both sides.

$$-13a^{2}+52a+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -13 for a, 52 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-52±\sqrt{52^{2}-4\left(-13\right)\times 24}}{2\left(-13\right)}$$

Square 52.

$$a=\frac{-52±\sqrt{2704-4\left(-13\right)\times 24}}{2\left(-13\right)}$$

Multiply -4 times -13.

$$a=\frac{-52±\sqrt{2704+52\times 24}}{2\left(-13\right)}$$

Multiply 52 times 24.

$$a=\frac{-52±\sqrt{2704+1248}}{2\left(-13\right)}$$

Add 2704 to 1248.

$$a=\frac{-52±\sqrt{3952}}{2\left(-13\right)}$$

Take the square root of 3952.

$$a=\frac{-52±4\sqrt{247}}{2\left(-13\right)}$$

Multiply 2 times -13.

$$a=\frac{-52±4\sqrt{247}}{-26}$$

Now solve the equation a=\frac{-52±4\sqrt{247}}{-26} when ± is plus. Add -52 to 4\sqrt{247}.

$$a=\frac{4\sqrt{247}-52}{-26}$$

Divide -52+4\sqrt{247} by -26.

$$a=-\frac{2\sqrt{247}}{13}+2$$

Now solve the equation a=\frac{-52±4\sqrt{247}}{-26} when ± is minus. Subtract 4\sqrt{247} from -52.

$$a=\frac{-4\sqrt{247}-52}{-26}$$

Divide -52-4\sqrt{247} by -26.

$$a=\frac{2\sqrt{247}}{13}+2$$

The equation is now solved.

$$a=-\frac{2\sqrt{247}}{13}+2 a=\frac{2\sqrt{247}}{13}+2$$

Steps for Completing the Square

Use the distributive property to multiply 8 by 2a^{2}-7a-3.

$$3a^{2}-4a=16a^{2}-56a-24$$

Subtract 16a^{2} from both sides.

$$3a^{2}-4a-16a^{2}=-56a-24$$

Combine 3a^{2} and -16a^{2} to get -13a^{2}.

$$-13a^{2}-4a=-56a-24$$

Add 56a to both sides.

$$-13a^{2}-4a+56a=-24$$

Combine -4a and 56a to get 52a.

$$-13a^{2}+52a=-24$$

Divide both sides by -13.

$$\frac{-13a^{2}+52a}{-13}=-\frac{24}{-13}$$

Dividing by -13 undoes the multiplication by -13.

$$a^{2}+\frac{52}{-13}a=-\frac{24}{-13}$$

Divide 52 by -13.

$$a^{2}-4a=-\frac{24}{-13}$$

Divide -24 by -13.

$$a^{2}-4a=\frac{24}{13}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-4a+\left(-2\right)^{2}=\frac{24}{13}+\left(-2\right)^{2}$$

Square -2.

$$a^{2}-4a+4=\frac{24}{13}+4$$

Add \frac{24}{13} to 4.

$$a^{2}-4a+4=\frac{76}{13}$$

Factor a^{2}-4a+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}.

$$\left(a-2\right)^{2}=\frac{76}{13}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-2\right)^{2}}=\sqrt{\frac{76}{13}}$$

Simplify.

$$a-2=\frac{2\sqrt{247}}{13} a-2=-\frac{2\sqrt{247}}{13}$$

Add 2 to both sides of the equation.

$$a=\frac{2\sqrt{247}}{13}+2 a=-\frac{2\sqrt{247}}{13}+2$$