Solve 6(8-x)(7-x)=1

$$6 ( 8 - x ) ( 7 - x ) = 1$$

Solve for x

$$x=-\frac{\sqrt{15}}{6}+\frac{15}{2}\approx 6.854502776$$

$$x=\frac{\sqrt{15}}{6}+\frac{15}{2}\approx 8.145497224$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6 by 8-x.

$$\left(48-6x\right)\left(7-x\right)=1$$

Use the distributive property to multiply 48-6x by 7-x and combine like terms.

$$336-90x+6x^{2}=1$$

Subtract 1 from both sides.

$$336-90x+6x^{2}-1=0$$

Subtract 1 from 336 to get 335.

$$335-90x+6x^{2}=0$$

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.

$$6x^{2}-90x+335=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -90 for b, and 335 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 6\times 335}}{2\times 6}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 6\times 335}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-90\right)±\sqrt{8100-24\times 335}}{2\times 6}$$

Multiply -24 times 335.

$$x=\frac{-\left(-90\right)±\sqrt{8100-8040}}{2\times 6}$$

Add 8100 to -8040.

$$x=\frac{-\left(-90\right)±\sqrt{60}}{2\times 6}$$

Take the square root of 60.

$$x=\frac{-\left(-90\right)±2\sqrt{15}}{2\times 6}$$

The opposite of -90 is 90.

$$x=\frac{90±2\sqrt{15}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{90±2\sqrt{15}}{12}$$

Now solve the equation x=\frac{90±2\sqrt{15}}{12} when ± is plus. Add 90 to 2\sqrt{15}.

$$x=\frac{2\sqrt{15}+90}{12}$$

Divide 90+2\sqrt{15} by 12.

$$x=\frac{\sqrt{15}}{6}+\frac{15}{2}$$

Now solve the equation x=\frac{90±2\sqrt{15}}{12} when ± is minus. Subtract 2\sqrt{15} from 90.

$$x=\frac{90-2\sqrt{15}}{12}$$

Divide 90-2\sqrt{15} by 12.

$$x=-\frac{\sqrt{15}}{6}+\frac{15}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{15}}{6}+\frac{15}{2} x=-\frac{\sqrt{15}}{6}+\frac{15}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 6 by 8-x.

$$\left(48-6x\right)\left(7-x\right)=1$$

Use the distributive property to multiply 48-6x by 7-x and combine like terms.

$$336-90x+6x^{2}=1$$

Subtract 336 from both sides.

$$-90x+6x^{2}=1-336$$

Subtract 336 from 1 to get -335.

$$-90x+6x^{2}=-335$$

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.

$$6x^{2}-90x=-335$$

Divide both sides by 6.

$$\frac{6x^{2}-90x}{6}=-\frac{335}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{90}{6}\right)x=-\frac{335}{6}$$

Divide -90 by 6.

$$x^{2}-15x=-\frac{335}{6}$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-15x+\left(-\frac{15}{2}\right)^{2}=-\frac{335}{6}+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-15x+\frac{225}{4}=-\frac{335}{6}+\frac{225}{4}$$

Add -\frac{335}{6} to \frac{225}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-15x+\frac{225}{4}=\frac{5}{12}$$

Factor x^{2}-15x+\frac{225}{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}.

$$\left(x-\frac{15}{2}\right)^{2}=\frac{5}{12}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{2}\right)^{2}}=\sqrt{\frac{5}{12}}$$

Simplify.

$$x-\frac{15}{2}=\frac{\sqrt{15}}{6} x-\frac{15}{2}=-\frac{\sqrt{15}}{6}$$

Add \frac{15}{2} to both sides of the equation.

$$x=\frac{\sqrt{15}}{6}+\frac{15}{2} x=-\frac{\sqrt{15}}{6}+\frac{15}{2}$$