Solve left(20-xright)left(30-2xright)=78*6

$$\left( 20-x \right) \left( 30-2x \right) = 78 \times 6$$

Solve for x

$$x=2$$

$$x=33$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-x by 30-2x and combine like terms.

$$600-70x+2x^{2}=78\times 6$$

Multiply 78 and 6 to get 468.

$$600-70x+2x^{2}=468$$

Subtract 468 from both sides.

$$600-70x+2x^{2}-468=0$$

Subtract 468 from 600 to get 132.

$$132-70x+2x^{2}=0$$

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.

$$2x^{2}-70x+132=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -70 for b, and 132 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-70\right)±\sqrt{\left(-70\right)^{2}-4\times 2\times 132}}{2\times 2}$$

Square -70.

$$x=\frac{-\left(-70\right)±\sqrt{4900-4\times 2\times 132}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-70\right)±\sqrt{4900-8\times 132}}{2\times 2}$$

Multiply -8 times 132.

$$x=\frac{-\left(-70\right)±\sqrt{4900-1056}}{2\times 2}$$

Add 4900 to -1056.

$$x=\frac{-\left(-70\right)±\sqrt{3844}}{2\times 2}$$

Take the square root of 3844.

$$x=\frac{-\left(-70\right)±62}{2\times 2}$$

The opposite of -70 is 70.

$$x=\frac{70±62}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{70±62}{4}$$

Now solve the equation x=\frac{70±62}{4} when ± is plus. Add 70 to 62.

$$x=\frac{132}{4}$$

Divide 132 by 4.

$$x=33$$

Now solve the equation x=\frac{70±62}{4} when ± is minus. Subtract 62 from 70.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

The equation is now solved.

$$x=33 x=2$$

Steps for Completing the Square

Use the distributive property to multiply 20-x by 30-2x and combine like terms.

$$600-70x+2x^{2}=78\times 6$$

Multiply 78 and 6 to get 468.

$$600-70x+2x^{2}=468$$

Subtract 600 from both sides.

$$-70x+2x^{2}=468-600$$

Subtract 600 from 468 to get -132.

$$-70x+2x^{2}=-132$$

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.

$$2x^{2}-70x=-132$$

Divide both sides by 2.

$$\frac{2x^{2}-70x}{2}=-\frac{132}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{70}{2}\right)x=-\frac{132}{2}$$

Divide -70 by 2.

$$x^{2}-35x=-\frac{132}{2}$$

Divide -132 by 2.

$$x^{2}-35x=-66$$

Divide -35, the coefficient of the x term, by 2 to get -\frac{35}{2}. Then add the square of -\frac{35}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-66+\left(-\frac{35}{2}\right)^{2}$$

Square -\frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-35x+\frac{1225}{4}=-66+\frac{1225}{4}$$

Add -66 to \frac{1225}{4}.

$$x^{2}-35x+\frac{1225}{4}=\frac{961}{4}$$

Factor x^{2}-35x+\frac{1225}{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}.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{961}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{961}{4}}$$

Simplify.

$$x-\frac{35}{2}=\frac{31}{2} x-\frac{35}{2}=-\frac{31}{2}$$

Add \frac{35}{2} to both sides of the equation.

$$x=33 x=2$$