Solve 631x^2-3520x+3703=0

$$631 { x }^{ 2 } -3520x+3703 = 0$$

Solve for x

$$x = \frac{13 \sqrt{4503} + 1760}{631} \approx 4.171722903$$

$$x = \frac{1760 - 13 \sqrt{4503}}{631} \approx 1.406724007$$

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.

$$631x^{2}-3520x+3703=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 631 for a, -3520 for b, and 3703 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3520\right)±\sqrt{\left(-3520\right)^{2}-4\times 631\times 3703}}{2\times 631}$$

Square -3520.

$$x=\frac{-\left(-3520\right)±\sqrt{12390400-4\times 631\times 3703}}{2\times 631}$$

Multiply -4 times 631.

$$x=\frac{-\left(-3520\right)±\sqrt{12390400-2524\times 3703}}{2\times 631}$$

Multiply -2524 times 3703.

$$x=\frac{-\left(-3520\right)±\sqrt{12390400-9346372}}{2\times 631}$$

Add 12390400 to -9346372.

$$x=\frac{-\left(-3520\right)±\sqrt{3044028}}{2\times 631}$$

Take the square root of 3044028.

$$x=\frac{-\left(-3520\right)±26\sqrt{4503}}{2\times 631}$$

The opposite of -3520 is 3520.

$$x=\frac{3520±26\sqrt{4503}}{2\times 631}$$

Multiply 2 times 631.

$$x=\frac{3520±26\sqrt{4503}}{1262}$$

Now solve the equation x=\frac{3520±26\sqrt{4503}}{1262} when ± is plus. Add 3520 to 26\sqrt{4503}.

$$x=\frac{26\sqrt{4503}+3520}{1262}$$

Divide 3520+26\sqrt{4503} by 1262.

$$x=\frac{13\sqrt{4503}+1760}{631}$$

Now solve the equation x=\frac{3520±26\sqrt{4503}}{1262} when ± is minus. Subtract 26\sqrt{4503} from 3520.

$$x=\frac{3520-26\sqrt{4503}}{1262}$$

Divide 3520-26\sqrt{4503} by 1262.

$$x=\frac{1760-13\sqrt{4503}}{631}$$

The equation is now solved.

$$x=\frac{13\sqrt{4503}+1760}{631} x=\frac{1760-13\sqrt{4503}}{631}$$

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.

$$631x^{2}-3520x+3703=0$$

Subtract 3703 from both sides of the equation.

$$631x^{2}-3520x+3703-3703=-3703$$

Subtracting 3703 from itself leaves 0.

$$631x^{2}-3520x=-3703$$

Divide both sides by 631.

$$\frac{631x^{2}-3520x}{631}=-\frac{3703}{631}$$

Dividing by 631 undoes the multiplication by 631.

$$x^{2}-\frac{3520}{631}x=-\frac{3703}{631}$$

Divide -\frac{3520}{631}, the coefficient of the x term, by 2 to get -\frac{1760}{631}. Then add the square of -\frac{1760}{631} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3520}{631}x+\left(-\frac{1760}{631}\right)^{2}=-\frac{3703}{631}+\left(-\frac{1760}{631}\right)^{2}$$

Square -\frac{1760}{631} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3520}{631}x+\frac{3097600}{398161}=-\frac{3703}{631}+\frac{3097600}{398161}$$

Add -\frac{3703}{631} to \frac{3097600}{398161} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3520}{631}x+\frac{3097600}{398161}=\frac{761007}{398161}$$

Factor x^{2}-\frac{3520}{631}x+\frac{3097600}{398161}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1760}{631}\right)^{2}=\frac{761007}{398161}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1760}{631}\right)^{2}}=\sqrt{\frac{761007}{398161}}$$

Simplify.

$$x-\frac{1760}{631}=\frac{13\sqrt{4503}}{631} x-\frac{1760}{631}=-\frac{13\sqrt{4503}}{631}$$

Add \frac{1760}{631} to both sides of the equation.

$$x=\frac{13\sqrt{4503}+1760}{631} x=\frac{1760-13\sqrt{4503}}{631}$$