Solve (20-5x)(6-x)=125*6

$$( 20 - 5 x ) ( 6 - x ) = 125 \times 6$$

Solve for x

$$x=\sqrt{151}+5\approx 17.288205727$$

$$x=5-\sqrt{151}\approx -7.288205727$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 20-5x by 6-x and combine like terms.

$$120-50x+5x^{2}=125\times 6$$

Multiply 125 and 6 to get 750.

$$120-50x+5x^{2}=750$$

Subtract 750 from both sides.

$$120-50x+5x^{2}-750=0$$

Subtract 750 from 120 to get -630.

$$-630-50x+5x^{2}=0$$

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.

$$5x^{2}-50x-630=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, -50 for b, and -630 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\times 5\left(-630\right)}}{2\times 5}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\times 5\left(-630\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-\left(-50\right)±\sqrt{2500-20\left(-630\right)}}{2\times 5}$$

Multiply -20 times -630.

$$x=\frac{-\left(-50\right)±\sqrt{2500+12600}}{2\times 5}$$

Add 2500 to 12600.

$$x=\frac{-\left(-50\right)±\sqrt{15100}}{2\times 5}$$

Take the square root of 15100.

$$x=\frac{-\left(-50\right)±10\sqrt{151}}{2\times 5}$$

The opposite of -50 is 50.

$$x=\frac{50±10\sqrt{151}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{50±10\sqrt{151}}{10}$$

Now solve the equation x=\frac{50±10\sqrt{151}}{10} when ± is plus. Add 50 to 10\sqrt{151}.

$$x=\frac{10\sqrt{151}+50}{10}$$

Divide 50+10\sqrt{151} by 10.

$$x=\sqrt{151}+5$$

Now solve the equation x=\frac{50±10\sqrt{151}}{10} when ± is minus. Subtract 10\sqrt{151} from 50.

$$x=\frac{50-10\sqrt{151}}{10}$$

Divide 50-10\sqrt{151} by 10.

$$x=5-\sqrt{151}$$

The equation is now solved.

$$x=\sqrt{151}+5 x=5-\sqrt{151}$$

Steps for Completing the Square

Use the distributive property to multiply 20-5x by 6-x and combine like terms.

$$120-50x+5x^{2}=125\times 6$$

Multiply 125 and 6 to get 750.

$$120-50x+5x^{2}=750$$

Subtract 120 from both sides.

$$-50x+5x^{2}=750-120$$

Subtract 120 from 750 to get 630.

$$-50x+5x^{2}=630$$

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.

$$5x^{2}-50x=630$$

Divide both sides by 5.

$$\frac{5x^{2}-50x}{5}=\frac{630}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\left(-\frac{50}{5}\right)x=\frac{630}{5}$$

Divide -50 by 5.

$$x^{2}-10x=\frac{630}{5}$$

Divide 630 by 5.

$$x^{2}-10x=126$$

Divide -10, the coefficient of the x term, by 2 to get -5. Then add the square of -5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-10x+\left(-5\right)^{2}=126+\left(-5\right)^{2}$$

Square -5.

$$x^{2}-10x+25=126+25$$

Add 126 to 25.

$$x^{2}-10x+25=151$$

Factor x^{2}-10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-5\right)^{2}=151$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{151}$$

Simplify.

$$x-5=\sqrt{151} x-5=-\sqrt{151}$$

Add 5 to both sides of the equation.

$$x=\sqrt{151}+5 x=5-\sqrt{151}$$