Solve 36=(17-x)(x-2)

$$36=(17-x)(x-2)$$

Solve for x

$$x=5$$

$$x=14$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 17-x by x-2 and combine like terms.

$$36=19x-34-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$19x-34-x^{2}=36$$

Subtract 36 from both sides.

$$19x-34-x^{2}-36=0$$

Subtract 36 from -34 to get -70.

$$19x-70-x^{2}=0$$

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.

$$-x^{2}+19x-70=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 19 for b, and -70 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-1\right)\left(-70\right)}}{2\left(-1\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-1\right)\left(-70\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-19±\sqrt{361+4\left(-70\right)}}{2\left(-1\right)}$$

Multiply 4 times -70.

$$x=\frac{-19±\sqrt{361-280}}{2\left(-1\right)}$$

Add 361 to -280.

$$x=\frac{-19±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$x=\frac{-19±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-19±9}{-2}$$

Now solve the equation x=\frac{-19±9}{-2} when ± is plus. Add -19 to 9.

$$x=-\frac{10}{-2}$$

Divide -10 by -2.

$$x=5$$

Now solve the equation x=\frac{-19±9}{-2} when ± is minus. Subtract 9 from -19.

$$x=-\frac{28}{-2}$$

Divide -28 by -2.

$$x=14$$

The equation is now solved.

$$x=5 x=14$$

Steps for Completing the Square

Use the distributive property to multiply 17-x by x-2 and combine like terms.

$$36=19x-34-x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$19x-34-x^{2}=36$$

Add 34 to both sides.

$$19x-x^{2}=36+34$$

Add 36 and 34 to get 70.

$$19x-x^{2}=70$$

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.

$$-x^{2}+19x=70$$

Divide both sides by -1.

$$\frac{-x^{2}+19x}{-1}=\frac{70}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{19}{-1}x=\frac{70}{-1}$$

Divide 19 by -1.

$$x^{2}-19x=\frac{70}{-1}$$

Divide 70 by -1.

$$x^{2}-19x=-70$$

Divide -19, the coefficient of the x term, by 2 to get -\frac{19}{2}. Then add the square of -\frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-19x+\left(-\frac{19}{2}\right)^{2}=-70+\left(-\frac{19}{2}\right)^{2}$$

Square -\frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-19x+\frac{361}{4}=-70+\frac{361}{4}$$

Add -70 to \frac{361}{4}.

$$x^{2}-19x+\frac{361}{4}=\frac{81}{4}$$

Factor x^{2}-19x+\frac{361}{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}.

$$\left(x-\frac{19}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{19}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$x-\frac{19}{2}=\frac{9}{2} x-\frac{19}{2}=-\frac{9}{2}$$

Add \frac{19}{2} to both sides of the equation.

$$x=14 x=5$$