Solve (9-x)(x-4)=3,x

$$( 9 - x ) ( x - 4 ) = 3 , x$$

Solve for x

$$x = \frac{\sqrt{13} + 13}{2} \approx 8.302775638$$

$$x = \frac{13 - \sqrt{13}}{2} \approx 4.697224362$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9-x by x-4 and combine like terms.

$$13x-36-x^{2}=3$$

Subtract 3 from both sides.

$$13x-36-x^{2}-3=0$$

Subtract 3 from -36 to get -39.

$$13x-39-x^{2}=0$$

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.

$$-x^{2}+13x-39=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 13 for b, and -39 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\left(-1\right)\left(-39\right)}}{2\left(-1\right)}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\left(-1\right)\left(-39\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-13±\sqrt{169+4\left(-39\right)}}{2\left(-1\right)}$$

Multiply 4 times -39.

$$x=\frac{-13±\sqrt{169-156}}{2\left(-1\right)}$$

Add 169 to -156.

$$x=\frac{-13±\sqrt{13}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-13±\sqrt{13}}{-2}$$

Now solve the equation x=\frac{-13±\sqrt{13}}{-2} when ± is plus. Add -13 to \sqrt{13}.

$$x=\frac{\sqrt{13}-13}{-2}$$

Divide -13+\sqrt{13} by -2.

$$x=\frac{13-\sqrt{13}}{2}$$

Now solve the equation x=\frac{-13±\sqrt{13}}{-2} when ± is minus. Subtract \sqrt{13} from -13.

$$x=\frac{-\sqrt{13}-13}{-2}$$

Divide -13-\sqrt{13} by -2.

$$x=\frac{\sqrt{13}+13}{2}$$

The equation is now solved.

$$x=\frac{13-\sqrt{13}}{2} x=\frac{\sqrt{13}+13}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 9-x by x-4 and combine like terms.

$$13x-36-x^{2}=3$$

Add 36 to both sides.

$$13x-x^{2}=3+36$$

Add 3 and 36 to get 39.

$$13x-x^{2}=39$$

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.

$$-x^{2}+13x=39$$

Divide both sides by -1.

$$\frac{-x^{2}+13x}{-1}=\frac{39}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{13}{-1}x=\frac{39}{-1}$$

Divide 13 by -1.

$$x^{2}-13x=\frac{39}{-1}$$

Divide 39 by -1.

$$x^{2}-13x=-39$$

Divide -13, the coefficient of the x term, by 2 to get -\frac{13}{2}. Then add the square of -\frac{13}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=-39+\left(-\frac{13}{2}\right)^{2}$$

Square -\frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-13x+\frac{169}{4}=-39+\frac{169}{4}$$

Add -39 to \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{13}{4}$$

Factor x^{2}-13x+\frac{169}{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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-\frac{13}{2}=\frac{\sqrt{13}}{2} x-\frac{13}{2}=-\frac{\sqrt{13}}{2}$$

Add \frac{13}{2} to both sides of the equation.

$$x=\frac{\sqrt{13}+13}{2} x=\frac{13-\sqrt{13}}{2}$$