Solve 19528x+62x^2+16=0

$$19528x+62 { x }^{ 2 } +16=0$$

Solve for x

$$x=\frac{2\sqrt{5958419}-4882}{31}\approx -0.000819338$$

$$x=\frac{-2\sqrt{5958419}-4882}{31}\approx -314.966922597$$

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.

$$62x^{2}+19528x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 62 for a, 19528 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-19528±\sqrt{19528^{2}-4\times 62\times 16}}{2\times 62}$$

Square 19528.

$$x=\frac{-19528±\sqrt{381342784-4\times 62\times 16}}{2\times 62}$$

Multiply -4 times 62.

$$x=\frac{-19528±\sqrt{381342784-248\times 16}}{2\times 62}$$

Multiply -248 times 16.

$$x=\frac{-19528±\sqrt{381342784-3968}}{2\times 62}$$

Add 381342784 to -3968.

$$x=\frac{-19528±\sqrt{381338816}}{2\times 62}$$

Take the square root of 381338816.

$$x=\frac{-19528±8\sqrt{5958419}}{2\times 62}$$

Multiply 2 times 62.

$$x=\frac{-19528±8\sqrt{5958419}}{124}$$

Now solve the equation x=\frac{-19528±8\sqrt{5958419}}{124} when ± is plus. Add -19528 to 8\sqrt{5958419}.

$$x=\frac{8\sqrt{5958419}-19528}{124}$$

Divide -19528+8\sqrt{5958419} by 124.

$$x=\frac{2\sqrt{5958419}-4882}{31}$$

Now solve the equation x=\frac{-19528±8\sqrt{5958419}}{124} when ± is minus. Subtract 8\sqrt{5958419} from -19528.

$$x=\frac{-8\sqrt{5958419}-19528}{124}$$

Divide -19528-8\sqrt{5958419} by 124.

$$x=\frac{-2\sqrt{5958419}-4882}{31}$$

The equation is now solved.

$$x=\frac{2\sqrt{5958419}-4882}{31} x=\frac{-2\sqrt{5958419}-4882}{31}$$

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.

$$62x^{2}+19528x+16=0$$

Subtract 16 from both sides of the equation.

$$62x^{2}+19528x+16-16=-16$$

Subtracting 16 from itself leaves 0.

$$62x^{2}+19528x=-16$$

Divide both sides by 62.

$$\frac{62x^{2}+19528x}{62}=-\frac{16}{62}$$

Dividing by 62 undoes the multiplication by 62.

$$x^{2}+\frac{19528}{62}x=-\frac{16}{62}$$

Reduce the fraction \frac{19528}{62} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{9764}{31}x=-\frac{16}{62}$$

Reduce the fraction \frac{-16}{62} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{9764}{31}x=-\frac{8}{31}$$

Divide \frac{9764}{31}, the coefficient of the x term, by 2 to get \frac{4882}{31}. Then add the square of \frac{4882}{31} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{9764}{31}x+\left(\frac{4882}{31}\right)^{2}=-\frac{8}{31}+\left(\frac{4882}{31}\right)^{2}$$

Square \frac{4882}{31} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{9764}{31}x+\frac{23833924}{961}=-\frac{8}{31}+\frac{23833924}{961}$$

Add -\frac{8}{31} to \frac{23833924}{961} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{9764}{31}x+\frac{23833924}{961}=\frac{23833676}{961}$$

Factor x^{2}+\frac{9764}{31}x+\frac{23833924}{961}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{4882}{31}\right)^{2}=\frac{23833676}{961}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{4882}{31}\right)^{2}}=\sqrt{\frac{23833676}{961}}$$

Simplify.

$$x+\frac{4882}{31}=\frac{2\sqrt{5958419}}{31} x+\frac{4882}{31}=-\frac{2\sqrt{5958419}}{31}$$

Subtract \frac{4882}{31} from both sides of the equation.

$$x=\frac{2\sqrt{5958419}-4882}{31} x=\frac{-2\sqrt{5958419}-4882}{31}$$