Solve 2d^2-5*19d+3=0

$$2 d ^ { 2 } - 5 \cdot 19 d + 3 = 0$$

Solve for d

$$d = \frac{\sqrt{9001} + 95}{4} \approx 47.46840003$$

$$d=\frac{95-\sqrt{9001}}{4}\approx 0.03159997$$

Steps Using the Quadratic Formula

Multiply 5 and 19 to get 95.

$$2d^{2}-95d+3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -95 for b, and 3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-95\right)±\sqrt{\left(-95\right)^{2}-4\times 2\times 3}}{2\times 2}$$

Square -95.

$$d=\frac{-\left(-95\right)±\sqrt{9025-4\times 2\times 3}}{2\times 2}$$

Multiply -4 times 2.

$$d=\frac{-\left(-95\right)±\sqrt{9025-8\times 3}}{2\times 2}$$

Multiply -8 times 3.

$$d=\frac{-\left(-95\right)±\sqrt{9025-24}}{2\times 2}$$

Add 9025 to -24.

$$d=\frac{-\left(-95\right)±\sqrt{9001}}{2\times 2}$$

The opposite of -95 is 95.

$$d=\frac{95±\sqrt{9001}}{2\times 2}$$

Multiply 2 times 2.

$$d=\frac{95±\sqrt{9001}}{4}$$

Now solve the equation d=\frac{95±\sqrt{9001}}{4} when ± is plus. Add 95 to \sqrt{9001}.

$$d=\frac{\sqrt{9001}+95}{4}$$

Now solve the equation d=\frac{95±\sqrt{9001}}{4} when ± is minus. Subtract \sqrt{9001} from 95.

$$d=\frac{95-\sqrt{9001}}{4}$$

The equation is now solved.

$$d=\frac{\sqrt{9001}+95}{4} d=\frac{95-\sqrt{9001}}{4}$$

Steps for Completing the Square

Multiply 5 and 19 to get 95.

$$2d^{2}-95d+3=0$$

Subtract 3 from both sides. Anything subtracted from zero gives its negation.

$$2d^{2}-95d=-3$$

Divide both sides by 2.

$$\frac{2d^{2}-95d}{2}=-\frac{3}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$d^{2}-\frac{95}{2}d=-\frac{3}{2}$$

Divide -\frac{95}{2}, the coefficient of the x term, by 2 to get -\frac{95}{4}. Then add the square of -\frac{95}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{95}{2}d+\left(-\frac{95}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{95}{4}\right)^{2}$$

Square -\frac{95}{4} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{95}{2}d+\frac{9025}{16}=-\frac{3}{2}+\frac{9025}{16}$$

Add -\frac{3}{2} to \frac{9025}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}-\frac{95}{2}d+\frac{9025}{16}=\frac{9001}{16}$$

Factor d^{2}-\frac{95}{2}d+\frac{9025}{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}.

$$\left(d-\frac{95}{4}\right)^{2}=\frac{9001}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{95}{4}\right)^{2}}=\sqrt{\frac{9001}{16}}$$

Simplify.

$$d-\frac{95}{4}=\frac{\sqrt{9001}}{4} d-\frac{95}{4}=-\frac{\sqrt{9001}}{4}$$

Add \frac{95}{4} to both sides of the equation.

$$d=\frac{\sqrt{9001}+95}{4} d=\frac{95-\sqrt{9001}}{4}$$