Solve x*(48-3x)=180

$$x \times (48-3x)=180$$

Solve for x

$$x=6$$

$$x=10$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by 48-3x.

$$48x-3x^{2}=180$$

Subtract 180 from both sides.

$$48x-3x^{2}-180=0$$

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.

$$-3x^{2}+48x-180=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 48 for b, and -180 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-48±\sqrt{48^{2}-4\left(-3\right)\left(-180\right)}}{2\left(-3\right)}$$

Square 48.

$$x=\frac{-48±\sqrt{2304-4\left(-3\right)\left(-180\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-48±\sqrt{2304+12\left(-180\right)}}{2\left(-3\right)}$$

Multiply 12 times -180.

$$x=\frac{-48±\sqrt{2304-2160}}{2\left(-3\right)}$$

Add 2304 to -2160.

$$x=\frac{-48±\sqrt{144}}{2\left(-3\right)}$$

Take the square root of 144.

$$x=\frac{-48±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-48±12}{-6}$$

Now solve the equation x=\frac{-48±12}{-6} when ± is plus. Add -48 to 12.

$$x=-\frac{36}{-6}$$

Divide -36 by -6.

$$x=6$$

Now solve the equation x=\frac{-48±12}{-6} when ± is minus. Subtract 12 from -48.

$$x=-\frac{60}{-6}$$

Divide -60 by -6.

$$x=10$$

The equation is now solved.

$$x=6 x=10$$

Steps for Completing the Square

Use the distributive property to multiply x by 48-3x.

$$48x-3x^{2}=180$$

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.

$$-3x^{2}+48x=180$$

Divide both sides by -3.

$$\frac{-3x^{2}+48x}{-3}=\frac{180}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{48}{-3}x=\frac{180}{-3}$$

Divide 48 by -3.

$$x^{2}-16x=\frac{180}{-3}$$

Divide 180 by -3.

$$x^{2}-16x=-60$$

Divide -16, the coefficient of the x term, by 2 to get -8. Then add the square of -8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-16x+\left(-8\right)^{2}=-60+\left(-8\right)^{2}$$

Square -8.

$$x^{2}-16x+64=-60+64$$

Add -60 to 64.

$$x^{2}-16x+64=4$$

Factor x^{2}-16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-8\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-8=2 x-8=-2$$

Add 8 to both sides of the equation.

$$x=10 x=6$$