Solve (38-3x)(2x)/4=84

$$\frac { ( 38 - 3 x ) ( 2 x ) } { 4 } = 84$$

Solve for x (complex solution)

$$x=\frac{-\sqrt{143}i+19}{3}\approx 6.333333333-3.986086914i$$

$$x=\frac{19+\sqrt{143}i}{3}\approx 6.333333333+3.986086914i$$

Steps Using the Quadratic Formula

Multiply both sides by 4.

$$\left(38-3x\right)\times 2x=84\times 4$$

Use the distributive property to multiply 38-3x by 2.

$$\left(76-6x\right)x=84\times 4$$

Use the distributive property to multiply 76-6x by x.

$$76x-6x^{2}=84\times 4$$

Multiply 84 and 4 to get 336.

$$76x-6x^{2}=336$$

Subtract 336 from both sides.

$$76x-6x^{2}-336=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-6x^{2}+76x-336=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 76 for b, and -336 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-76±\sqrt{76^{2}-4\left(-6\right)\left(-336\right)}}{2\left(-6\right)}$$

Square 76.

$$x=\frac{-76±\sqrt{5776-4\left(-6\right)\left(-336\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-76±\sqrt{5776+24\left(-336\right)}}{2\left(-6\right)}$$

Multiply 24 times -336.

$$x=\frac{-76±\sqrt{5776-8064}}{2\left(-6\right)}$$

Add 5776 to -8064.

$$x=\frac{-76±\sqrt{-2288}}{2\left(-6\right)}$$

Take the square root of -2288.

$$x=\frac{-76±4\sqrt{143}i}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-76±4\sqrt{143}i}{-12}$$

Now solve the equation x=\frac{-76±4\sqrt{143}i}{-12} when ± is plus. Add -76 to 4i\sqrt{143}.

$$x=\frac{-76+4\sqrt{143}i}{-12}$$

Divide -76+4i\sqrt{143} by -12.

$$x=\frac{-\sqrt{143}i+19}{3}$$

Now solve the equation x=\frac{-76±4\sqrt{143}i}{-12} when ± is minus. Subtract 4i\sqrt{143} from -76.

$$x=\frac{-4\sqrt{143}i-76}{-12}$$

Divide -76-4i\sqrt{143} by -12.

$$x=\frac{19+\sqrt{143}i}{3}$$

The equation is now solved.

$$x=\frac{-\sqrt{143}i+19}{3} x=\frac{19+\sqrt{143}i}{3}$$

Steps for Completing the Square

Multiply both sides by 4.

$$\left(38-3x\right)\times 2x=84\times 4$$

Use the distributive property to multiply 38-3x by 2.

$$\left(76-6x\right)x=84\times 4$$

Use the distributive property to multiply 76-6x by x.

$$76x-6x^{2}=84\times 4$$

Multiply 84 and 4 to get 336.

$$76x-6x^{2}=336$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-6x^{2}+76x=336$$

Divide both sides by -6.

$$\frac{-6x^{2}+76x}{-6}=\frac{336}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{76}{-6}x=\frac{336}{-6}$$

Reduce the fraction \frac{76}{-6} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{38}{3}x=\frac{336}{-6}$$

Divide 336 by -6.

$$x^{2}-\frac{38}{3}x=-56$$

Divide -\frac{38}{3}, the coefficient of the x term, by 2 to get -\frac{19}{3}. Then add the square of -\frac{19}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{38}{3}x+\left(-\frac{19}{3}\right)^{2}=-56+\left(-\frac{19}{3}\right)^{2}$$

Square -\frac{19}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=-56+\frac{361}{9}$$

Add -56 to \frac{361}{9}.

$$x^{2}-\frac{38}{3}x+\frac{361}{9}=-\frac{143}{9}$$

Factor x^{2}-\frac{38}{3}x+\frac{361}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{19}{3}\right)^{2}=-\frac{143}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{3}\right)^{2}}=\sqrt{-\frac{143}{9}}$$

Simplify.

$$x-\frac{19}{3}=\frac{\sqrt{143}i}{3} x-\frac{19}{3}=-\frac{\sqrt{143}i}{3}$$

Add \frac{19}{3} to both sides of the equation.

$$x=\frac{19+\sqrt{143}i}{3} x=\frac{-\sqrt{143}i+19}{3}$$