Solve xleft(x-7right)/3-1=x/2-x-4/3

$$\frac{ x \left( x-7 \right) }{ 3 } -1 = \frac{ x }{ 2 } - \frac{ x-4 }{ 3 }$$

Solve for x

$$x = \frac{\sqrt{337} + 15}{4} \approx 8.339389938$$

$$x=\frac{15-\sqrt{337}}{4}\approx -0.839389938$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2x\left(x-7\right)-6=3x-2\left(x-4\right)$$

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

$$2x^{2}-14x-6=3x-2\left(x-4\right)$$

Use the distributive property to multiply -2 by x-4.

$$2x^{2}-14x-6=3x-2x+8$$

Combine 3x and -2x to get x.

$$2x^{2}-14x-6=x+8$$

Subtract x from both sides.

$$2x^{2}-14x-6-x=8$$

Combine -14x and -x to get -15x.

$$2x^{2}-15x-6=8$$

Subtract 8 from both sides.

$$2x^{2}-15x-6-8=0$$

Subtract 8 from -6 to get -14.

$$2x^{2}-15x-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -15 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 2\left(-14\right)}}{2\times 2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 2\left(-14\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-15\right)±\sqrt{225-8\left(-14\right)}}{2\times 2}$$

Multiply -8 times -14.

$$x=\frac{-\left(-15\right)±\sqrt{225+112}}{2\times 2}$$

Add 225 to 112.

$$x=\frac{-\left(-15\right)±\sqrt{337}}{2\times 2}$$

The opposite of -15 is 15.

$$x=\frac{15±\sqrt{337}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{15±\sqrt{337}}{4}$$

Now solve the equation x=\frac{15±\sqrt{337}}{4} when ± is plus. Add 15 to \sqrt{337}.

$$x=\frac{\sqrt{337}+15}{4}$$

Now solve the equation x=\frac{15±\sqrt{337}}{4} when ± is minus. Subtract \sqrt{337} from 15.

$$x=\frac{15-\sqrt{337}}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{337}+15}{4} x=\frac{15-\sqrt{337}}{4}$$

Steps for Completing the Square

Multiply both sides of the equation by 6, the least common multiple of 3,2.

$$2x\left(x-7\right)-6=3x-2\left(x-4\right)$$

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

$$2x^{2}-14x-6=3x-2\left(x-4\right)$$

Use the distributive property to multiply -2 by x-4.

$$2x^{2}-14x-6=3x-2x+8$$

Combine 3x and -2x to get x.

$$2x^{2}-14x-6=x+8$$

Subtract x from both sides.

$$2x^{2}-14x-6-x=8$$

Combine -14x and -x to get -15x.

$$2x^{2}-15x-6=8$$

Add 6 to both sides.

$$2x^{2}-15x=8+6$$

Add 8 and 6 to get 14.

$$2x^{2}-15x=14$$

Divide both sides by 2.

$$\frac{2x^{2}-15x}{2}=\frac{14}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}-\frac{15}{2}x=\frac{14}{2}$$

Divide 14 by 2.

$$x^{2}-\frac{15}{2}x=7$$

Divide -\frac{15}{2}, the coefficient of the x term, by 2 to get -\frac{15}{4}. Then add the square of -\frac{15}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{2}x+\left(-\frac{15}{4}\right)^{2}=7+\left(-\frac{15}{4}\right)^{2}$$

Square -\frac{15}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=7+\frac{225}{16}$$

Add 7 to \frac{225}{16}.

$$x^{2}-\frac{15}{2}x+\frac{225}{16}=\frac{337}{16}$$

Factor x^{2}-\frac{15}{2}x+\frac{225}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{4}\right)^{2}=\frac{337}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{4}\right)^{2}}=\sqrt{\frac{337}{16}}$$

Simplify.

$$x-\frac{15}{4}=\frac{\sqrt{337}}{4} x-\frac{15}{4}=-\frac{\sqrt{337}}{4}$$

Add \frac{15}{4} to both sides of the equation.

$$x=\frac{\sqrt{337}+15}{4} x=\frac{15-\sqrt{337}}{4}$$