Solve (x^2-x+8):4=(4x-6):3+4

$$( x ^ { 2 } - x + 8 ) : 4 = ( 4 x - 6 ) : 3 + 4$$

Solve for x

$$x = \frac{19}{3} = 6\frac{1}{3} \approx 6.333333333$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x^{2}-x+8\right)=4\left(4x-6\right)+48$$

Use the distributive property to multiply 3 by x^{2}-x+8.

$$3x^{2}-3x+24=4\left(4x-6\right)+48$$

Use the distributive property to multiply 4 by 4x-6.

$$3x^{2}-3x+24=16x-24+48$$

Add -24 and 48 to get 24.

$$3x^{2}-3x+24=16x+24$$

Subtract 16x from both sides.

$$3x^{2}-3x+24-16x=24$$

Combine -3x and -16x to get -19x.

$$3x^{2}-19x+24=24$$

Subtract 24 from both sides.

$$3x^{2}-19x+24-24=0$$

Subtract 24 from 24 to get 0.

$$3x^{2}-19x=0$$

Factor out x.

$$x\left(3x-19\right)=0$$

To find equation solutions, solve x=0 and 3x-19=0.

$$x=0 x=\frac{19}{3}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x^{2}-x+8\right)=4\left(4x-6\right)+48$$

Use the distributive property to multiply 3 by x^{2}-x+8.

$$3x^{2}-3x+24=4\left(4x-6\right)+48$$

Use the distributive property to multiply 4 by 4x-6.

$$3x^{2}-3x+24=16x-24+48$$

Add -24 and 48 to get 24.

$$3x^{2}-3x+24=16x+24$$

Subtract 16x from both sides.

$$3x^{2}-3x+24-16x=24$$

Combine -3x and -16x to get -19x.

$$3x^{2}-19x+24=24$$

Subtract 24 from both sides.

$$3x^{2}-19x+24-24=0$$

Subtract 24 from 24 to get 0.

$$3x^{2}-19x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -19 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}}}{2\times 3}$$

Take the square root of \left(-19\right)^{2}.

$$x=\frac{-\left(-19\right)±19}{2\times 3}$$

The opposite of -19 is 19.

$$x=\frac{19±19}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{19±19}{6}$$

Now solve the equation x=\frac{19±19}{6} when ± is plus. Add 19 to 19.

$$x=\frac{38}{6}$$

Reduce the fraction \frac{38}{6} to lowest terms by extracting and canceling out 2.

$$x=\frac{19}{3}$$

Now solve the equation x=\frac{19±19}{6} when ± is minus. Subtract 19 from 19.

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

Divide 0 by 6.

$$x=0$$

The equation is now solved.

$$x=\frac{19}{3} x=0$$

Steps for Completing the Square

Multiply both sides of the equation by 12, the least common multiple of 4,3.

$$3\left(x^{2}-x+8\right)=4\left(4x-6\right)+48$$

Use the distributive property to multiply 3 by x^{2}-x+8.

$$3x^{2}-3x+24=4\left(4x-6\right)+48$$

Use the distributive property to multiply 4 by 4x-6.

$$3x^{2}-3x+24=16x-24+48$$

Add -24 and 48 to get 24.

$$3x^{2}-3x+24=16x+24$$

Subtract 16x from both sides.

$$3x^{2}-3x+24-16x=24$$

Combine -3x and -16x to get -19x.

$$3x^{2}-19x+24=24$$

Subtract 24 from both sides.

$$3x^{2}-19x=24-24$$

Subtract 24 from 24 to get 0.

$$3x^{2}-19x=0$$

Divide both sides by 3.

$$\frac{3x^{2}-19x}{3}=\frac{0}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}-\frac{19}{3}x=\frac{0}{3}$$

Divide 0 by 3.

$$x^{2}-\frac{19}{3}x=0$$

Divide -\frac{19}{3}, the coefficient of the x term, by 2 to get -\frac{19}{6}. Then add the square of -\frac{19}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{19}{3}x+\left(-\frac{19}{6}\right)^{2}=\left(-\frac{19}{6}\right)^{2}$$

Square -\frac{19}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{19}{3}x+\frac{361}{36}=\frac{361}{36}$$

Factor x^{2}-\frac{19}{3}x+\frac{361}{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}.

$$\left(x-\frac{19}{6}\right)^{2}=\frac{361}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{6}\right)^{2}}=\sqrt{\frac{361}{36}}$$

Simplify.

$$x-\frac{19}{6}=\frac{19}{6} x-\frac{19}{6}=-\frac{19}{6}$$

Add \frac{19}{6} to both sides of the equation.

$$x=\frac{19}{3} x=0$$