Solve (66-x)*x=56*(-x+(66-x))

$$(66-x) \times x=56 \times (-x+(66-x))$$

Solve for x

$$x=24$$

$$x=154$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 66-x by x.

$$66x-x^{2}=56\left(-x+66-x\right)$$

Use the distributive property to multiply 56 by -x+66-x.

$$66x-x^{2}=56\left(-x\right)+3696-56x$$

Subtract 56\left(-x\right) from both sides.

$$66x-x^{2}-56\left(-x\right)=3696-56x$$

Subtract 3696 from both sides.

$$66x-x^{2}-56\left(-x\right)-3696=-56x$$

Add 56x to both sides.

$$66x-x^{2}-56\left(-x\right)-3696+56x=0$$

Multiply -1 and 56 to get -56.

$$66x-x^{2}-56\left(-1\right)x-3696+56x=0$$

Multiply -56 and -1 to get 56.

$$66x-x^{2}+56x-3696+56x=0$$

Combine 66x and 56x to get 122x.

$$122x-x^{2}-3696+56x=0$$

Combine 122x and 56x to get 178x.

$$178x-x^{2}-3696=0$$

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.

$$-x^{2}+178x-3696=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 178 for b, and -3696 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-178±\sqrt{178^{2}-4\left(-1\right)\left(-3696\right)}}{2\left(-1\right)}$$

Square 178.

$$x=\frac{-178±\sqrt{31684-4\left(-1\right)\left(-3696\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-178±\sqrt{31684+4\left(-3696\right)}}{2\left(-1\right)}$$

Multiply 4 times -3696.

$$x=\frac{-178±\sqrt{31684-14784}}{2\left(-1\right)}$$

Add 31684 to -14784.

$$x=\frac{-178±\sqrt{16900}}{2\left(-1\right)}$$

Take the square root of 16900.

$$x=\frac{-178±130}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-178±130}{-2}$$

Now solve the equation x=\frac{-178±130}{-2} when ± is plus. Add -178 to 130.

$$x=-\frac{48}{-2}$$

Divide -48 by -2.

$$x=24$$

Now solve the equation x=\frac{-178±130}{-2} when ± is minus. Subtract 130 from -178.

$$x=-\frac{308}{-2}$$

Divide -308 by -2.

$$x=154$$

The equation is now solved.

$$x=24 x=154$$

Steps for Completing the Square

Use the distributive property to multiply 66-x by x.

$$66x-x^{2}=56\left(-x+66-x\right)$$

Use the distributive property to multiply 56 by -x+66-x.

$$66x-x^{2}=56\left(-x\right)+3696-56x$$

Subtract 56\left(-x\right) from both sides.

$$66x-x^{2}-56\left(-x\right)=3696-56x$$

Add 56x to both sides.

$$66x-x^{2}-56\left(-x\right)+56x=3696$$

Multiply -1 and 56 to get -56.

$$66x-x^{2}-56\left(-1\right)x+56x=3696$$

Multiply -56 and -1 to get 56.

$$66x-x^{2}+56x+56x=3696$$

Combine 66x and 56x to get 122x.

$$122x-x^{2}+56x=3696$$

Combine 122x and 56x to get 178x.

$$178x-x^{2}=3696$$

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.

$$-x^{2}+178x=3696$$

Divide both sides by -1.

$$\frac{-x^{2}+178x}{-1}=\frac{3696}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{178}{-1}x=\frac{3696}{-1}$$

Divide 178 by -1.

$$x^{2}-178x=\frac{3696}{-1}$$

Divide 3696 by -1.

$$x^{2}-178x=-3696$$

Divide -178, the coefficient of the x term, by 2 to get -89. Then add the square of -89 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-178x+\left(-89\right)^{2}=-3696+\left(-89\right)^{2}$$

Square -89.

$$x^{2}-178x+7921=-3696+7921$$

Add -3696 to 7921.

$$x^{2}-178x+7921=4225$$

Factor x^{2}-178x+7921. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-89\right)^{2}=4225$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-89\right)^{2}}=\sqrt{4225}$$

Simplify.

$$x-89=65 x-89=-65$$

Add 89 to both sides of the equation.

$$x=154 x=24$$