Solve 4x(x+39)+9=0

$$4x(x+39)+9=0$$

Solve for x

$$x=3\sqrt{42}-\frac{39}{2}\approx -0.057777905$$

$$x=-3\sqrt{42}-\frac{39}{2}\approx -38.942222095$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+39.

$$4x^{2}+156x+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 156 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-156±\sqrt{156^{2}-4\times 4\times 9}}{2\times 4}$$

Square 156.

$$x=\frac{-156±\sqrt{24336-4\times 4\times 9}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-156±\sqrt{24336-16\times 9}}{2\times 4}$$

Multiply -16 times 9.

$$x=\frac{-156±\sqrt{24336-144}}{2\times 4}$$

Add 24336 to -144.

$$x=\frac{-156±\sqrt{24192}}{2\times 4}$$

Take the square root of 24192.

$$x=\frac{-156±24\sqrt{42}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-156±24\sqrt{42}}{8}$$

Now solve the equation x=\frac{-156±24\sqrt{42}}{8} when ± is plus. Add -156 to 24\sqrt{42}.

$$x=\frac{24\sqrt{42}-156}{8}$$

Divide -156+24\sqrt{42} by 8.

$$x=3\sqrt{42}-\frac{39}{2}$$

Now solve the equation x=\frac{-156±24\sqrt{42}}{8} when ± is minus. Subtract 24\sqrt{42} from -156.

$$x=\frac{-24\sqrt{42}-156}{8}$$

Divide -156-24\sqrt{42} by 8.

$$x=-3\sqrt{42}-\frac{39}{2}$$

The equation is now solved.

$$x=3\sqrt{42}-\frac{39}{2} x=-3\sqrt{42}-\frac{39}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+39.

$$4x^{2}+156x+9=0$$

Subtract 9 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}+156x=-9$$

Divide both sides by 4.

$$\frac{4x^{2}+156x}{4}=-\frac{9}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{156}{4}x=-\frac{9}{4}$$

Divide 156 by 4.

$$x^{2}+39x=-\frac{9}{4}$$

Divide 39, the coefficient of the x term, by 2 to get \frac{39}{2}. Then add the square of \frac{39}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+39x+\left(\frac{39}{2}\right)^{2}=-\frac{9}{4}+\left(\frac{39}{2}\right)^{2}$$

Square \frac{39}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+39x+\frac{1521}{4}=\frac{-9+1521}{4}$$

Add -\frac{9}{4} to \frac{1521}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+39x+\frac{1521}{4}=378$$

Factor x^{2}+39x+\frac{1521}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{39}{2}\right)^{2}=378$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{39}{2}\right)^{2}}=\sqrt{378}$$

Simplify.

$$x+\frac{39}{2}=3\sqrt{42} x+\frac{39}{2}=-3\sqrt{42}$$

Subtract \frac{39}{2} from both sides of the equation.

$$x=3\sqrt{42}-\frac{39}{2} x=-3\sqrt{42}-\frac{39}{2}$$