Solve (4-x)(3x)=9/2

$$( 4 - x ) ( 3 x ) = \frac { 9 } { 2 }$$

Solve for x

$$x=\frac{\sqrt{10}}{2}+2\approx 3.58113883$$

$$x=-\frac{\sqrt{10}}{2}+2\approx 0.41886117$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4-x by 3.

$$\left(12-3x\right)x=\frac{9}{2}$$

Use the distributive property to multiply 12-3x by x.

$$12x-3x^{2}=\frac{9}{2}$$

Subtract \frac{9}{2} from both sides.

$$12x-3x^{2}-\frac{9}{2}=0$$

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.

$$-3x^{2}+12x-\frac{9}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 12 for b, and -\frac{9}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-3\right)\left(-\frac{9}{2}\right)}}{2\left(-3\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-3\right)\left(-\frac{9}{2}\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-12±\sqrt{144+12\left(-\frac{9}{2}\right)}}{2\left(-3\right)}$$

Multiply 12 times -\frac{9}{2}.

$$x=\frac{-12±\sqrt{144-54}}{2\left(-3\right)}$$

Add 144 to -54.

$$x=\frac{-12±\sqrt{90}}{2\left(-3\right)}$$

Take the square root of 90.

$$x=\frac{-12±3\sqrt{10}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-12±3\sqrt{10}}{-6}$$

Now solve the equation x=\frac{-12±3\sqrt{10}}{-6} when ± is plus. Add -12 to 3\sqrt{10}.

$$x=\frac{3\sqrt{10}-12}{-6}$$

Divide -12+3\sqrt{10} by -6.

$$x=-\frac{\sqrt{10}}{2}+2$$

Now solve the equation x=\frac{-12±3\sqrt{10}}{-6} when ± is minus. Subtract 3\sqrt{10} from -12.

$$x=\frac{-3\sqrt{10}-12}{-6}$$

Divide -12-3\sqrt{10} by -6.

$$x=\frac{\sqrt{10}}{2}+2$$

The equation is now solved.

$$x=-\frac{\sqrt{10}}{2}+2 x=\frac{\sqrt{10}}{2}+2$$

Steps for Completing the Square

Use the distributive property to multiply 4-x by 3.

$$\left(12-3x\right)x=\frac{9}{2}$$

Use the distributive property to multiply 12-3x by x.

$$12x-3x^{2}=\frac{9}{2}$$

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.

$$-3x^{2}+12x=\frac{9}{2}$$

Divide both sides by -3.

$$\frac{-3x^{2}+12x}{-3}=\frac{\frac{9}{2}}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{12}{-3}x=\frac{\frac{9}{2}}{-3}$$

Divide 12 by -3.

$$x^{2}-4x=\frac{\frac{9}{2}}{-3}$$

Divide \frac{9}{2} by -3.

$$x^{2}-4x=-\frac{3}{2}$$

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.

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

Square -2.

$$x^{2}-4x+4=-\frac{3}{2}+4$$

Add -\frac{3}{2} to 4.

$$x^{2}-4x+4=\frac{5}{2}$$

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

$$\left(x-2\right)^{2}=\frac{5}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{5}{2}}$$

Simplify.

$$x-2=\frac{\sqrt{10}}{2} x-2=-\frac{\sqrt{10}}{2}$$

Add 2 to both sides of the equation.

$$x=\frac{\sqrt{10}}{2}+2 x=-\frac{\sqrt{10}}{2}+2$$