Solve (k-4)(k-1)=18

$$( k - 4 ) ( k - 1 ) = 18$$

Solve for k

$$k=-2$$

$$k=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply k-4 by k-1 and combine like terms.

$$k^{2}-5k+4=18$$

Subtract 18 from both sides.

$$k^{2}-5k+4-18=0$$

Subtract 18 from 4 to get -14.

$$k^{2}-5k-14=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -5 for b, and -14 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-14\right)}}{2}$$

Square -5.

$$k=\frac{-\left(-5\right)±\sqrt{25-4\left(-14\right)}}{2}$$

Multiply -4 times -14.

$$k=\frac{-\left(-5\right)±\sqrt{25+56}}{2}$$

Add 25 to 56.

$$k=\frac{-\left(-5\right)±\sqrt{81}}{2}$$

Take the square root of 81.

$$k=\frac{-\left(-5\right)±9}{2}$$

The opposite of -5 is 5.

$$k=\frac{5±9}{2}$$

Now solve the equation k=\frac{5±9}{2} when ± is plus. Add 5 to 9.

$$k=\frac{14}{2}$$

Divide 14 by 2.

$$k=7$$

Now solve the equation k=\frac{5±9}{2} when ± is minus. Subtract 9 from 5.

$$k=-\frac{4}{2}$$

Divide -4 by 2.

$$k=-2$$

The equation is now solved.

$$k=7 k=-2$$

Steps for Completing the Square

Use the distributive property to multiply k-4 by k-1 and combine like terms.

$$k^{2}-5k+4=18$$

Subtract 4 from both sides.

$$k^{2}-5k=18-4$$

Subtract 4 from 18 to get 14.

$$k^{2}-5k=14$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-5k+\left(-\frac{5}{2}\right)^{2}=14+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-5k+\frac{25}{4}=14+\frac{25}{4}$$

Add 14 to \frac{25}{4}.

$$k^{2}-5k+\frac{25}{4}=\frac{81}{4}$$

Factor k^{2}-5k+\frac{25}{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}.

$$\left(k-\frac{5}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{5}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$k-\frac{5}{2}=\frac{9}{2} k-\frac{5}{2}=-\frac{9}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$k=7 k=-2$$