Solve 4470x^2-4355x+0.765=0

$$4470 x ^ { 2 } - 4355 x + 0.765 = 0$$

Solve for x

$$x=\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}\approx 0.974097239$$

$$x=-\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}\approx 0.000175692$$

Steps Using the Quadratic Formula

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.

$$4470x^{2}-4355x+0.765=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4470 for a, -4355 for b, and 0.765 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4355\right)±\sqrt{\left(-4355\right)^{2}-4\times 4470\times 0.765}}{2\times 4470}$$

Square -4355.

$$x=\frac{-\left(-4355\right)±\sqrt{18966025-4\times 4470\times 0.765}}{2\times 4470}$$

Multiply -4 times 4470.

$$x=\frac{-\left(-4355\right)±\sqrt{18966025-17880\times 0.765}}{2\times 4470}$$

Multiply -17880 times 0.765.

$$x=\frac{-\left(-4355\right)±\sqrt{18966025-13678.2}}{2\times 4470}$$

Add 18966025 to -13678.2.

$$x=\frac{-\left(-4355\right)±\sqrt{18952346.8}}{2\times 4470}$$

Take the square root of 18952346.8.

$$x=\frac{-\left(-4355\right)±\frac{\sqrt{473808670}}{5}}{2\times 4470}$$

The opposite of -4355 is 4355.

$$x=\frac{4355±\frac{\sqrt{473808670}}{5}}{2\times 4470}$$

Multiply 2 times 4470.

$$x=\frac{4355±\frac{\sqrt{473808670}}{5}}{8940}$$

Now solve the equation x=\frac{4355±\frac{\sqrt{473808670}}{5}}{8940} when ± is plus. Add 4355 to \frac{\sqrt{473808670}}{5}.

$$x=\frac{\frac{\sqrt{473808670}}{5}+4355}{8940}$$

Divide 4355+\frac{\sqrt{473808670}}{5} by 8940.

$$x=\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}$$

Now solve the equation x=\frac{4355±\frac{\sqrt{473808670}}{5}}{8940} when ± is minus. Subtract \frac{\sqrt{473808670}}{5} from 4355.

$$x=\frac{-\frac{\sqrt{473808670}}{5}+4355}{8940}$$

Divide 4355-\frac{\sqrt{473808670}}{5} by 8940.

$$x=-\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}$$

The equation is now solved.

$$x=\frac{\sqrt{473808670}}{44700}+\frac{871}{1788} x=-\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}$$

Steps for Completing the Square

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.

$$4470x^{2}-4355x+0.765=0$$

Subtract 0.765 from both sides of the equation.

$$4470x^{2}-4355x+0.765-0.765=-0.765$$

Subtracting 0.765 from itself leaves 0.

$$4470x^{2}-4355x=-0.765$$

Divide both sides by 4470.

$$\frac{4470x^{2}-4355x}{4470}=-\frac{0.765}{4470}$$

Dividing by 4470 undoes the multiplication by 4470.

$$x^{2}+\left(-\frac{4355}{4470}\right)x=-\frac{0.765}{4470}$$

Reduce the fraction \frac{-4355}{4470} to lowest terms by extracting and canceling out 5.

$$x^{2}-\frac{871}{894}x=-\frac{0.765}{4470}$$

Divide -0.765 by 4470.

$$x^{2}-\frac{871}{894}x=-\frac{51}{298000}$$

Divide -\frac{871}{894}, the coefficient of the x term, by 2 to get -\frac{871}{1788}. Then add the square of -\frac{871}{1788} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{871}{894}x+\left(-\frac{871}{1788}\right)^{2}=-\frac{51}{298000}+\left(-\frac{871}{1788}\right)^{2}$$

Square -\frac{871}{1788} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{871}{894}x+\frac{758641}{3196944}=-\frac{51}{298000}+\frac{758641}{3196944}$$

Add -\frac{51}{298000} to \frac{758641}{3196944} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{871}{894}x+\frac{758641}{3196944}=\frac{47380867}{199809000}$$

Factor x^{2}-\frac{871}{894}x+\frac{758641}{3196944}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{871}{1788}\right)^{2}=\frac{47380867}{199809000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{871}{1788}\right)^{2}}=\sqrt{\frac{47380867}{199809000}}$$

Simplify.

$$x-\frac{871}{1788}=\frac{\sqrt{473808670}}{44700} x-\frac{871}{1788}=-\frac{\sqrt{473808670}}{44700}$$

Add \frac{871}{1788} to both sides of the equation.

$$x=\frac{\sqrt{473808670}}{44700}+\frac{871}{1788} x=-\frac{\sqrt{473808670}}{44700}+\frac{871}{1788}$$