Solve left(4-dright)left(4+5dright)=14

$$\left( 4-d \right) \left( 4+5d \right) = 14$$

Solve for d

$$d = \frac{\sqrt{74} + 8}{5} \approx 3.320465053$$

$$d=\frac{8-\sqrt{74}}{5}\approx -0.120465053$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4-d by 4+5d and combine like terms.

$$16+16d-5d^{2}=14$$

Subtract 14 from both sides.

$$16+16d-5d^{2}-14=0$$

Subtract 14 from 16 to get 2.

$$2+16d-5d^{2}=0$$

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.

$$-5d^{2}+16d+2=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 16 for b, and 2 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-16±\sqrt{16^{2}-4\left(-5\right)\times 2}}{2\left(-5\right)}$$

Square 16.

$$d=\frac{-16±\sqrt{256-4\left(-5\right)\times 2}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$d=\frac{-16±\sqrt{256+20\times 2}}{2\left(-5\right)}$$

Multiply 20 times 2.

$$d=\frac{-16±\sqrt{256+40}}{2\left(-5\right)}$$

Add 256 to 40.

$$d=\frac{-16±\sqrt{296}}{2\left(-5\right)}$$

Take the square root of 296.

$$d=\frac{-16±2\sqrt{74}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$d=\frac{-16±2\sqrt{74}}{-10}$$

Now solve the equation d=\frac{-16±2\sqrt{74}}{-10} when ± is plus. Add -16 to 2\sqrt{74}.

$$d=\frac{2\sqrt{74}-16}{-10}$$

Divide -16+2\sqrt{74} by -10.

$$d=\frac{8-\sqrt{74}}{5}$$

Now solve the equation d=\frac{-16±2\sqrt{74}}{-10} when ± is minus. Subtract 2\sqrt{74} from -16.

$$d=\frac{-2\sqrt{74}-16}{-10}$$

Divide -16-2\sqrt{74} by -10.

$$d=\frac{\sqrt{74}+8}{5}$$

The equation is now solved.

$$d=\frac{8-\sqrt{74}}{5} d=\frac{\sqrt{74}+8}{5}$$

Steps for Completing the Square

Use the distributive property to multiply 4-d by 4+5d and combine like terms.

$$16+16d-5d^{2}=14$$

Subtract 16 from both sides.

$$16d-5d^{2}=14-16$$

Subtract 16 from 14 to get -2.

$$16d-5d^{2}=-2$$

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.

$$-5d^{2}+16d=-2$$

Divide both sides by -5.

$$\frac{-5d^{2}+16d}{-5}=-\frac{2}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$d^{2}+\frac{16}{-5}d=-\frac{2}{-5}$$

Divide 16 by -5.

$$d^{2}-\frac{16}{5}d=-\frac{2}{-5}$$

Divide -2 by -5.

$$d^{2}-\frac{16}{5}d=\frac{2}{5}$$

Divide -\frac{16}{5}, the coefficient of the x term, by 2 to get -\frac{8}{5}. Then add the square of -\frac{8}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{16}{5}d+\left(-\frac{8}{5}\right)^{2}=\frac{2}{5}+\left(-\frac{8}{5}\right)^{2}$$

Square -\frac{8}{5} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{16}{5}d+\frac{64}{25}=\frac{2}{5}+\frac{64}{25}$$

Add \frac{2}{5} to \frac{64}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}-\frac{16}{5}d+\frac{64}{25}=\frac{74}{25}$$

Factor d^{2}-\frac{16}{5}d+\frac{64}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{8}{5}\right)^{2}=\frac{74}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{8}{5}\right)^{2}}=\sqrt{\frac{74}{25}}$$

Simplify.

$$d-\frac{8}{5}=\frac{\sqrt{74}}{5} d-\frac{8}{5}=-\frac{\sqrt{74}}{5}$$

Add \frac{8}{5} to both sides of the equation.

$$d=\frac{\sqrt{74}+8}{5} d=\frac{8-\sqrt{74}}{5}$$