Solve (x-30)(162-3x)=420

$$(x-30)(162-3x)=420$$

Solve for x

$$x=40$$

$$x=44$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-30 by 162-3x and combine like terms.

$$252x-3x^{2}-4860=420$$

Subtract 420 from both sides.

$$252x-3x^{2}-4860-420=0$$

Subtract 420 from -4860 to get -5280.

$$252x-3x^{2}-5280=0$$

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.

$$-3x^{2}+252x-5280=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 252 for b, and -5280 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-252±\sqrt{252^{2}-4\left(-3\right)\left(-5280\right)}}{2\left(-3\right)}$$

Square 252.

$$x=\frac{-252±\sqrt{63504-4\left(-3\right)\left(-5280\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-252±\sqrt{63504+12\left(-5280\right)}}{2\left(-3\right)}$$

Multiply 12 times -5280.

$$x=\frac{-252±\sqrt{63504-63360}}{2\left(-3\right)}$$

Add 63504 to -63360.

$$x=\frac{-252±\sqrt{144}}{2\left(-3\right)}$$

Take the square root of 144.

$$x=\frac{-252±12}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-252±12}{-6}$$

Now solve the equation x=\frac{-252±12}{-6} when ± is plus. Add -252 to 12.

$$x=-\frac{240}{-6}$$

Divide -240 by -6.

$$x=40$$

Now solve the equation x=\frac{-252±12}{-6} when ± is minus. Subtract 12 from -252.

$$x=-\frac{264}{-6}$$

Divide -264 by -6.

$$x=44$$

The equation is now solved.

$$x=40 x=44$$

Steps for Completing the Square

Use the distributive property to multiply x-30 by 162-3x and combine like terms.

$$252x-3x^{2}-4860=420$$

Add 4860 to both sides.

$$252x-3x^{2}=420+4860$$

Add 420 and 4860 to get 5280.

$$252x-3x^{2}=5280$$

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.

$$-3x^{2}+252x=5280$$

Divide both sides by -3.

$$\frac{-3x^{2}+252x}{-3}=\frac{5280}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{252}{-3}x=\frac{5280}{-3}$$

Divide 252 by -3.

$$x^{2}-84x=\frac{5280}{-3}$$

Divide 5280 by -3.

$$x^{2}-84x=-1760$$

Divide -84, the coefficient of the x term, by 2 to get -42. Then add the square of -42 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-84x+\left(-42\right)^{2}=-1760+\left(-42\right)^{2}$$

Square -42.

$$x^{2}-84x+1764=-1760+1764$$

Add -1760 to 1764.

$$x^{2}-84x+1764=4$$

Factor x^{2}-84x+1764. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-42\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-42\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-42=2 x-42=-2$$

Add 42 to both sides of the equation.

$$x=44 x=40$$