Solve (10-x)*(4-x)=27

$$(10-x) \times (4-x)=27$$

Solve for x

$$x=13$$

$$x=1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 10-x by 4-x and combine like terms.

$$40-14x+x^{2}=27$$

Subtract 27 from both sides.

$$40-14x+x^{2}-27=0$$

Subtract 27 from 40 to get 13.

$$13-14x+x^{2}=0$$

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.

$$x^{2}-14x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -14 for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 13}}{2}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 13}}{2}$$

Multiply -4 times 13.

$$x=\frac{-\left(-14\right)±\sqrt{196-52}}{2}$$

Add 196 to -52.

$$x=\frac{-\left(-14\right)±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-\left(-14\right)±12}{2}$$

The opposite of -14 is 14.

$$x=\frac{14±12}{2}$$

Now solve the equation x=\frac{14±12}{2} when ± is plus. Add 14 to 12.

$$x=\frac{26}{2}$$

Divide 26 by 2.

$$x=13$$

Now solve the equation x=\frac{14±12}{2} when ± is minus. Subtract 12 from 14.

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

Divide 2 by 2.

$$x=1$$

The equation is now solved.

$$x=13 x=1$$

Steps for Completing the Square

Use the distributive property to multiply 10-x by 4-x and combine like terms.

$$40-14x+x^{2}=27$$

Subtract 40 from both sides.

$$-14x+x^{2}=27-40$$

Subtract 40 from 27 to get -13.

$$-14x+x^{2}=-13$$

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.

$$x^{2}-14x=-13$$

Divide -14, the coefficient of the x term, by 2 to get -7. Then add the square of -7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-14x+\left(-7\right)^{2}=-13+\left(-7\right)^{2}$$

Square -7.

$$x^{2}-14x+49=-13+49$$

Add -13 to 49.

$$x^{2}-14x+49=36$$

Factor x^{2}-14x+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-7\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{36}$$

Simplify.

$$x-7=6 x-7=-6$$

Add 7 to both sides of the equation.

$$x=13 x=1$$