Solve (48+2k)(48-3k)=18

$$( 48 + 2 k ) ( 48 - 3 k ) = 18$$

Solve for k

$$k=\sqrt{397}-4\approx 15.924858845$$

$$k=-\sqrt{397}-4\approx -23.924858845$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 48+2k by 48-3k and combine like terms.

$$2304-48k-6k^{2}=18$$

Subtract 18 from both sides.

$$2304-48k-6k^{2}-18=0$$

Subtract 18 from 2304 to get 2286.

$$2286-48k-6k^{2}=0$$

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.

$$-6k^{2}-48k+2286=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -6 for a, -48 for b, and 2286 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-6\right)\times 2286}}{2\left(-6\right)}$$

Square -48.

$$k=\frac{-\left(-48\right)±\sqrt{2304-4\left(-6\right)\times 2286}}{2\left(-6\right)}$$

Multiply -4 times -6.

$$k=\frac{-\left(-48\right)±\sqrt{2304+24\times 2286}}{2\left(-6\right)}$$

Multiply 24 times 2286.

$$k=\frac{-\left(-48\right)±\sqrt{2304+54864}}{2\left(-6\right)}$$

Add 2304 to 54864.

$$k=\frac{-\left(-48\right)±\sqrt{57168}}{2\left(-6\right)}$$

Take the square root of 57168.

$$k=\frac{-\left(-48\right)±12\sqrt{397}}{2\left(-6\right)}$$

The opposite of -48 is 48.

$$k=\frac{48±12\sqrt{397}}{2\left(-6\right)}$$

Multiply 2 times -6.

$$k=\frac{48±12\sqrt{397}}{-12}$$

Now solve the equation k=\frac{48±12\sqrt{397}}{-12} when ± is plus. Add 48 to 12\sqrt{397}.

$$k=\frac{12\sqrt{397}+48}{-12}$$

Divide 48+12\sqrt{397} by -12.

$$k=-\left(\sqrt{397}+4\right)$$

Now solve the equation k=\frac{48±12\sqrt{397}}{-12} when ± is minus. Subtract 12\sqrt{397} from 48.

$$k=\frac{48-12\sqrt{397}}{-12}$$

Divide 48-12\sqrt{397} by -12.

$$k=\sqrt{397}-4$$

The equation is now solved.

$$k=-\left(\sqrt{397}+4\right) k=\sqrt{397}-4$$

Steps for Completing the Square

Use the distributive property to multiply 48+2k by 48-3k and combine like terms.

$$2304-48k-6k^{2}=18$$

Subtract 2304 from both sides.

$$-48k-6k^{2}=18-2304$$

Subtract 2304 from 18 to get -2286.

$$-48k-6k^{2}=-2286$$

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.

$$-6k^{2}-48k=-2286$$

Divide both sides by -6.

$$\frac{-6k^{2}-48k}{-6}=-\frac{2286}{-6}$$

Dividing by -6 undoes the multiplication by -6.

$$k^{2}+\left(-\frac{48}{-6}\right)k=-\frac{2286}{-6}$$

Divide -48 by -6.

$$k^{2}+8k=-\frac{2286}{-6}$$

Divide -2286 by -6.

$$k^{2}+8k=381$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}+8k+4^{2}=381+4^{2}$$

Square 4.

$$k^{2}+8k+16=381+16$$

Add 381 to 16.

$$k^{2}+8k+16=397$$

Factor k^{2}+8k+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k+4\right)^{2}=397$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k+4\right)^{2}}=\sqrt{397}$$

Simplify.

$$k+4=\sqrt{397} k+4=-\sqrt{397}$$

Subtract 4 from both sides of the equation.

$$k=\sqrt{397}-4 k=-\sqrt{397}-4$$