Solve (14-x)(6x-24)/10=12.6

$$\frac{ (14-x)(6x-24) }{ 10 } =12.6$$

Solve for x

$$x=7$$

$$x=11$$

Steps Using the Quadratic Formula

Multiply both sides by 10.

$$\left(14-x\right)\left(6x-24\right)=12.6\times 10$$

Use the distributive property to multiply 14-x by 6x-24 and combine like terms.

$$108x-336-6x^{2}=12.6\times 10$$

Multiply 12.6 and 10 to get 126.

$$108x-336-6x^{2}=126$$

Subtract 126 from both sides.

$$108x-336-6x^{2}-126=0$$

Subtract 126 from -336 to get -462.

$$108x-462-6x^{2}=0$$

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.

$$-6x^{2}+108x-462=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, 108 for b, and -462 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-108±\sqrt{108^{2}-4\left(-6\right)\left(-462\right)}}{2\left(-6\right)}$$

Square 108.

$$x=\frac{-108±\sqrt{11664-4\left(-6\right)\left(-462\right)}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$x=\frac{-108±\sqrt{11664+24\left(-462\right)}}{2\left(-6\right)}$$

Multiply 24 times -462.

$$x=\frac{-108±\sqrt{11664-11088}}{2\left(-6\right)}$$

Add 11664 to -11088.

$$x=\frac{-108±\sqrt{576}}{2\left(-6\right)}$$

Take the square root of 576.

$$x=\frac{-108±24}{2\left(-6\right)}$$

Multiply 2 times -6.

$$x=\frac{-108±24}{-12}$$

Now solve the equation x=\frac{-108±24}{-12} when ± is plus. Add -108 to 24.

$$x=-\frac{84}{-12}$$

Divide -84 by -12.

$$x=7$$

Now solve the equation x=\frac{-108±24}{-12} when ± is minus. Subtract 24 from -108.

$$x=-\frac{132}{-12}$$

Divide -132 by -12.

$$x=11$$

The equation is now solved.

$$x=7 x=11$$

Steps for Completing the Square

Multiply both sides by 10.

$$\left(14-x\right)\left(6x-24\right)=12.6\times 10$$

Use the distributive property to multiply 14-x by 6x-24 and combine like terms.

$$108x-336-6x^{2}=12.6\times 10$$

Multiply 12.6 and 10 to get 126.

$$108x-336-6x^{2}=126$$

Add 336 to both sides.

$$108x-6x^{2}=126+336$$

Add 126 and 336 to get 462.

$$108x-6x^{2}=462$$

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.

$$-6x^{2}+108x=462$$

Divide both sides by -6.

$$\frac{-6x^{2}+108x}{-6}=\frac{462}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$x^{2}+\frac{108}{-6}x=\frac{462}{-6}$$

Divide 108 by -6.

$$x^{2}-18x=\frac{462}{-6}$$

Divide 462 by -6.

$$x^{2}-18x=-77$$

Divide -18, the coefficient of the x term, by 2 to get -9. Then add the square of -9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-18x+\left(-9\right)^{2}=-77+\left(-9\right)^{2}$$

Square -9.

$$x^{2}-18x+81=-77+81$$

Add -77 to 81.

$$x^{2}-18x+81=4$$

Factor x^{2}-18x+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-9\right)^{2}=4$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-9\right)^{2}}=\sqrt{4}$$

Simplify.

$$x-9=2 x-9=-2$$

Add 9 to both sides of the equation.

$$x=11 x=7$$