Solve 6x*(9-2x)=54

$$6 x \cdot ( 9 - 2 x ) = 54$$

Solve for x

$$x=3$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x by 9-2x.

$$54x-12x^{2}=54$$

Subtract 54 from both sides.

$$54x-12x^{2}-54=0$$

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.

$$-12x^{2}+54x-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -12 for a, 54 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-54±\sqrt{54^{2}-4\left(-12\right)\left(-54\right)}}{2\left(-12\right)}$$

Square 54.

$$x=\frac{-54±\sqrt{2916-4\left(-12\right)\left(-54\right)}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$x=\frac{-54±\sqrt{2916+48\left(-54\right)}}{2\left(-12\right)}$$

Multiply 48 times -54.

$$x=\frac{-54±\sqrt{2916-2592}}{2\left(-12\right)}$$

Add 2916 to -2592.

$$x=\frac{-54±\sqrt{324}}{2\left(-12\right)}$$

Take the square root of 324.

$$x=\frac{-54±18}{2\left(-12\right)}$$

Multiply 2 times -12.

$$x=\frac{-54±18}{-24}$$

Now solve the equation x=\frac{-54±18}{-24} when ± is plus. Add -54 to 18.

$$x=-\frac{36}{-24}$$

Reduce the fraction \frac{-36}{-24} to lowest terms by extracting and canceling out 12.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{-54±18}{-24} when ± is minus. Subtract 18 from -54.

$$x=-\frac{72}{-24}$$

Divide -72 by -24.

$$x=3$$

The equation is now solved.

$$x=\frac{3}{2} x=3$$

Steps for Completing the Square

Use the distributive property to multiply 6x by 9-2x.

$$54x-12x^{2}=54$$

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.

$$-12x^{2}+54x=54$$

Divide both sides by -12.

$$\frac{-12x^{2}+54x}{-12}=\frac{54}{-12}$$

Dividing by -12 undoes the multiplication by -12.

$$x^{2}+\frac{54}{-12}x=\frac{54}{-12}$$

Reduce the fraction \frac{54}{-12} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{9}{2}x=\frac{54}{-12}$$

Reduce the fraction \frac{54}{-12} to lowest terms by extracting and canceling out 6.

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

Divide -\frac{9}{2}, the coefficient of the x term, by 2 to get -\frac{9}{4}. Then add the square of -\frac{9}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=-\frac{9}{2}+\left(-\frac{9}{4}\right)^{2}$$

Square -\frac{9}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{9}{2}+\frac{81}{16}$$

Add -\frac{9}{2} to \frac{81}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{9}{16}$$

Factor x^{2}-\frac{9}{2}x+\frac{81}{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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{9}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Simplify.

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

Add \frac{9}{4} to both sides of the equation.

$$x=3 x=\frac{3}{2}$$