Solve d(7d-6)+4=0b

$$d ( 7 d - 6 ) + 4 = 0 b$$

Solve for d

$$d=\frac{3+\sqrt{19}i}{7}$$

$$d=\frac{-\sqrt{19}i+3}{7}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply d by 7d-6.

$$7d^{2}-6d+4=0b$$

Anything times zero gives zero.

$$7d^{2}-6d+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -6 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$d=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 7\times 4}}{2\times 7}$$

Square -6.

$$d=\frac{-\left(-6\right)±\sqrt{36-4\times 7\times 4}}{2\times 7}$$

Multiply -4 times 7.

$$d=\frac{-\left(-6\right)±\sqrt{36-28\times 4}}{2\times 7}$$

Multiply -28 times 4.

$$d=\frac{-\left(-6\right)±\sqrt{36-112}}{2\times 7}$$

Add 36 to -112.

$$d=\frac{-\left(-6\right)±\sqrt{-76}}{2\times 7}$$

Take the square root of -76.

$$d=\frac{-\left(-6\right)±2\sqrt{19}i}{2\times 7}$$

The opposite of -6 is 6.

$$d=\frac{6±2\sqrt{19}i}{2\times 7}$$

Multiply 2 times 7.

$$d=\frac{6±2\sqrt{19}i}{14}$$

Now solve the equation d=\frac{6±2\sqrt{19}i}{14} when ± is plus. Add 6 to 2i\sqrt{19}.

$$d=\frac{6+2\sqrt{19}i}{14}$$

Divide 6+2i\sqrt{19} by 14.

$$d=\frac{3+\sqrt{19}i}{7}$$

Now solve the equation d=\frac{6±2\sqrt{19}i}{14} when ± is minus. Subtract 2i\sqrt{19} from 6.

$$d=\frac{-2\sqrt{19}i+6}{14}$$

Divide 6-2i\sqrt{19} by 14.

$$d=\frac{-\sqrt{19}i+3}{7}$$

The equation is now solved.

$$d=\frac{3+\sqrt{19}i}{7} d=\frac{-\sqrt{19}i+3}{7}$$

Steps for Completing the Square

Use the distributive property to multiply d by 7d-6.

$$7d^{2}-6d+4=0b$$

Anything times zero gives zero.

$$7d^{2}-6d+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$7d^{2}-6d=-4$$

Divide both sides by 7.

$$\frac{7d^{2}-6d}{7}=-\frac{4}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$d^{2}-\frac{6}{7}d=-\frac{4}{7}$$

Divide -\frac{6}{7}, the coefficient of the x term, by 2 to get -\frac{3}{7}. Then add the square of -\frac{3}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$d^{2}-\frac{6}{7}d+\left(-\frac{3}{7}\right)^{2}=-\frac{4}{7}+\left(-\frac{3}{7}\right)^{2}$$

Square -\frac{3}{7} by squaring both the numerator and the denominator of the fraction.

$$d^{2}-\frac{6}{7}d+\frac{9}{49}=-\frac{4}{7}+\frac{9}{49}$$

Add -\frac{4}{7} to \frac{9}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$d^{2}-\frac{6}{7}d+\frac{9}{49}=-\frac{19}{49}$$

Factor d^{2}-\frac{6}{7}d+\frac{9}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(d-\frac{3}{7}\right)^{2}=-\frac{19}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(d-\frac{3}{7}\right)^{2}}=\sqrt{-\frac{19}{49}}$$

Simplify.

$$d-\frac{3}{7}=\frac{\sqrt{19}i}{7} d-\frac{3}{7}=-\frac{\sqrt{19}i}{7}$$

Add \frac{3}{7} to both sides of the equation.

$$d=\frac{3+\sqrt{19}i}{7} d=\frac{-\sqrt{19}i+3}{7}$$

Solve for b

$$b\in \mathrm{C}$$

$$d=\frac{3+\sqrt{19}i}{7}\text{ or }d=\frac{-\sqrt{19}i+3}{7}$$