Solve [5-21x(1-x)+(2-4x)]=50

$$[ 5 - 21 x ( 1 - x ) + ( 2 - 4 x ) ] = 50$$

Solve for x

$$x = \frac{\sqrt{4237} + 25}{42} \approx 2.145053387$$

$$x=\frac{25-\sqrt{4237}}{42}\approx -0.954577196$$

Steps Using the Quadratic Formula

Subtract 50 from both sides.

$$5-21x\left(1-x\right)+2-4x-50=0$$

Subtract 50 from 2 to get -48.

$$5-21x\left(1-x\right)-48-4x=0$$

Add 48 to both sides. Anything plus zero gives itself.

$$5-21x\left(1-x\right)-4x=48$$

Subtract 48 from both sides.

$$5-21x\left(1-x\right)-4x-48=0$$

Use the distributive property to multiply -21x by 1-x.

$$5-21x+21x^{2}-4x-48=0$$

Combine -21x and -4x to get -25x.

$$5-25x+21x^{2}-48=0$$

Subtract 48 from 5 to get -43.

$$-43-25x+21x^{2}=0$$

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.

$$21x^{2}-25x-43=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 21 for a, -25 for b, and -43 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 21\left(-43\right)}}{2\times 21}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 21\left(-43\right)}}{2\times 21}$$

Multiply -4 times 21.

$$x=\frac{-\left(-25\right)±\sqrt{625-84\left(-43\right)}}{2\times 21}$$

Multiply -84 times -43.

$$x=\frac{-\left(-25\right)±\sqrt{625+3612}}{2\times 21}$$

Add 625 to 3612.

$$x=\frac{-\left(-25\right)±\sqrt{4237}}{2\times 21}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{4237}}{2\times 21}$$

Multiply 2 times 21.

$$x=\frac{25±\sqrt{4237}}{42}$$

Now solve the equation x=\frac{25±\sqrt{4237}}{42} when ± is plus. Add 25 to \sqrt{4237}.

$$x=\frac{\sqrt{4237}+25}{42}$$

Now solve the equation x=\frac{25±\sqrt{4237}}{42} when ± is minus. Subtract \sqrt{4237} from 25.

$$x=\frac{25-\sqrt{4237}}{42}$$

The equation is now solved.

$$x=\frac{\sqrt{4237}+25}{42} x=\frac{25-\sqrt{4237}}{42}$$

Steps for Completing the Square

Subtract 2 from both sides.

$$5-21x\left(1-x\right)-4x=50-2$$

Subtract 2 from 50 to get 48.

$$5-21x\left(1-x\right)-4x=48$$

Use the distributive property to multiply -21x by 1-x.

$$5-21x+21x^{2}-4x=48$$

Combine -21x and -4x to get -25x.

$$5-25x+21x^{2}=48$$

Subtract 5 from both sides.

$$-25x+21x^{2}=48-5$$

Subtract 5 from 48 to get 43.

$$-25x+21x^{2}=43$$

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.

$$21x^{2}-25x=43$$

Divide both sides by 21.

$$\frac{21x^{2}-25x}{21}=\frac{43}{21}$$

Dividing by 21 undoes the multiplication by 21.

$$x^{2}-\frac{25}{21}x=\frac{43}{21}$$

Divide -\frac{25}{21}, the coefficient of the x term, by 2 to get -\frac{25}{42}. Then add the square of -\frac{25}{42} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{21}x+\left(-\frac{25}{42}\right)^{2}=\frac{43}{21}+\left(-\frac{25}{42}\right)^{2}$$

Square -\frac{25}{42} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{21}x+\frac{625}{1764}=\frac{43}{21}+\frac{625}{1764}$$

Add \frac{43}{21} to \frac{625}{1764} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{21}x+\frac{625}{1764}=\frac{4237}{1764}$$

Factor x^{2}-\frac{25}{21}x+\frac{625}{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}.

$$\left(x-\frac{25}{42}\right)^{2}=\frac{4237}{1764}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{42}\right)^{2}}=\sqrt{\frac{4237}{1764}}$$

Simplify.

$$x-\frac{25}{42}=\frac{\sqrt{4237}}{42} x-\frac{25}{42}=-\frac{\sqrt{4237}}{42}$$

Add \frac{25}{42} to both sides of the equation.

$$x=\frac{\sqrt{4237}+25}{42} x=\frac{25-\sqrt{4237}}{42}$$