Solve 2473*4x^2+14x-1*4=0

$$2473 \cdot 4 x ^ { 2 } + 14 x - 1 \cdot 4 = 0$$

Solve for x

$$x=\frac{\sqrt{39617}-7}{9892}\approx 0.019413688$$

$$x=\frac{-\sqrt{39617}-7}{9892}\approx -0.020828973$$

Steps Using the Quadratic Formula

Multiply 2473 and 4 to get 9892.

$$9892x^{2}+14x-1\times 4=0$$

Multiply 1 and 4 to get 4.

$$9892x^{2}+14x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9892 for a, 14 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 9892\left(-4\right)}}{2\times 9892}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 9892\left(-4\right)}}{2\times 9892}$$

Multiply -4 times 9892.

$$x=\frac{-14±\sqrt{196-39568\left(-4\right)}}{2\times 9892}$$

Multiply -39568 times -4.

$$x=\frac{-14±\sqrt{196+158272}}{2\times 9892}$$

Add 196 to 158272.

$$x=\frac{-14±\sqrt{158468}}{2\times 9892}$$

Take the square root of 158468.

$$x=\frac{-14±2\sqrt{39617}}{2\times 9892}$$

Multiply 2 times 9892.

$$x=\frac{-14±2\sqrt{39617}}{19784}$$

Now solve the equation x=\frac{-14±2\sqrt{39617}}{19784} when ± is plus. Add -14 to 2\sqrt{39617}.

$$x=\frac{2\sqrt{39617}-14}{19784}$$

Divide -14+2\sqrt{39617} by 19784.

$$x=\frac{\sqrt{39617}-7}{9892}$$

Now solve the equation x=\frac{-14±2\sqrt{39617}}{19784} when ± is minus. Subtract 2\sqrt{39617} from -14.

$$x=\frac{-2\sqrt{39617}-14}{19784}$$

Divide -14-2\sqrt{39617} by 19784.

$$x=\frac{-\sqrt{39617}-7}{9892}$$

The equation is now solved.

$$x=\frac{\sqrt{39617}-7}{9892} x=\frac{-\sqrt{39617}-7}{9892}$$

Steps for Completing the Square

Multiply 2473 and 4 to get 9892.

$$9892x^{2}+14x-1\times 4=0$$

Multiply 1 and 4 to get 4.

$$9892x^{2}+14x-4=0$$

Add 4 to both sides. Anything plus zero gives itself.

$$9892x^{2}+14x=4$$

Divide both sides by 9892.

$$\frac{9892x^{2}+14x}{9892}=\frac{4}{9892}$$

Dividing by 9892 undoes the multiplication by 9892.

$$x^{2}+\frac{14}{9892}x=\frac{4}{9892}$$

Reduce the fraction \frac{14}{9892} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{4946}x=\frac{4}{9892}$$

Reduce the fraction \frac{4}{9892} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{7}{4946}x=\frac{1}{2473}$$

Divide \frac{7}{4946}, the coefficient of the x term, by 2 to get \frac{7}{9892}. Then add the square of \frac{7}{9892} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4946}x+\left(\frac{7}{9892}\right)^{2}=\frac{1}{2473}+\left(\frac{7}{9892}\right)^{2}$$

Square \frac{7}{9892} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4946}x+\frac{49}{97851664}=\frac{1}{2473}+\frac{49}{97851664}$$

Add \frac{1}{2473} to \frac{49}{97851664} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{7}{4946}x+\frac{49}{97851664}=\frac{39617}{97851664}$$

Factor x^{2}+\frac{7}{4946}x+\frac{49}{97851664}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{9892}\right)^{2}=\frac{39617}{97851664}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{9892}\right)^{2}}=\sqrt{\frac{39617}{97851664}}$$

Simplify.

$$x+\frac{7}{9892}=\frac{\sqrt{39617}}{9892} x+\frac{7}{9892}=-\frac{\sqrt{39617}}{9892}$$

Subtract \frac{7}{9892} from both sides of the equation.

$$x=\frac{\sqrt{39617}-7}{9892} x=\frac{-\sqrt{39617}-7}{9892}$$