Solve 4d^2=5d+2

$$4 d ^ { 2 } = 5 d + 2$$

Solve for d

$$d = \frac{\sqrt{57} + 5}{8} \approx 1.568729304$$

$$d=\frac{5-\sqrt{57}}{8}\approx -0.318729304$$

Steps Using the Quadratic Formula

Subtract 5d from both sides.

$$4d^{2}-5d=2$$

Subtract 2 from both sides.

$$4d^{2}-5d-2=0$$

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

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

Square -5.

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

Multiply -4 times 4.

$$d=\frac{-\left(-5\right)±\sqrt{25-16\left(-2\right)}}{2\times 4}$$

Multiply -16 times -2.

$$d=\frac{-\left(-5\right)±\sqrt{25+32}}{2\times 4}$$

Add 25 to 32.

$$d=\frac{-\left(-5\right)±\sqrt{57}}{2\times 4}$$

The opposite of -5 is 5.

$$d=\frac{5±\sqrt{57}}{2\times 4}$$

Multiply 2 times 4.

$$d=\frac{5±\sqrt{57}}{8}$$

Now solve the equation d=\frac{5±\sqrt{57}}{8} when ± is plus. Add 5 to \sqrt{57}.

$$d=\frac{\sqrt{57}+5}{8}$$

Now solve the equation d=\frac{5±\sqrt{57}}{8} when ± is minus. Subtract \sqrt{57} from 5.

$$d=\frac{5-\sqrt{57}}{8}$$

The equation is now solved.

$$d=\frac{\sqrt{57}+5}{8} d=\frac{5-\sqrt{57}}{8}$$

Steps for Completing the Square

Subtract 5d from both sides.

$$4d^{2}-5d=2$$

Divide both sides by 4.

$$\frac{4d^{2}-5d}{4}=\frac{2}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$d^{2}-\frac{5}{4}d=\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$d^{2}-\frac{5}{4}d=\frac{1}{2}$$

Divide -\frac{5}{4}, the coefficient of the x term, by 2 to get -\frac{5}{8}. Then add the square of -\frac{5}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{5}{4}d+\left(-\frac{5}{8}\right)^{2}=\frac{1}{2}+\left(-\frac{5}{8}\right)^{2}$$

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

$$d^{2}-\frac{5}{4}d+\frac{25}{64}=\frac{1}{2}+\frac{25}{64}$$

Add \frac{1}{2} to \frac{25}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}-\frac{5}{4}d+\frac{25}{64}=\frac{57}{64}$$

Factor d^{2}-\frac{5}{4}d+\frac{25}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{5}{8}\right)^{2}=\frac{57}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{5}{8}\right)^{2}}=\sqrt{\frac{57}{64}}$$

Simplify.

$$d-\frac{5}{8}=\frac{\sqrt{57}}{8} d-\frac{5}{8}=-\frac{\sqrt{57}}{8}$$

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

$$d=\frac{\sqrt{57}+5}{8} d=\frac{5-\sqrt{57}}{8}$$