Solve 2448x^2+1376x+208=0

$$2448 { x }^{ 2 } +1376x+208 = 0$$

Solve for x (complex solution)

$$x=\frac{-43+2\sqrt{35}i}{153}\approx -0.281045752+0.077334376i$$

$$x=\frac{-2\sqrt{35}i-43}{153}\approx -0.281045752-0.077334376i$$

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.

$$2448x^{2}+1376x+208=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2448 for a, 1376 for b, and 208 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1376±\sqrt{1376^{2}-4\times 2448\times 208}}{2\times 2448}$$

Square 1376.

$$x=\frac{-1376±\sqrt{1893376-4\times 2448\times 208}}{2\times 2448}$$

Multiply -4 times 2448.

$$x=\frac{-1376±\sqrt{1893376-9792\times 208}}{2\times 2448}$$

Multiply -9792 times 208.

$$x=\frac{-1376±\sqrt{1893376-2036736}}{2\times 2448}$$

Add 1893376 to -2036736.

$$x=\frac{-1376±\sqrt{-143360}}{2\times 2448}$$

Take the square root of -143360.

$$x=\frac{-1376±64\sqrt{35}i}{2\times 2448}$$

Multiply 2 times 2448.

$$x=\frac{-1376±64\sqrt{35}i}{4896}$$

Now solve the equation x=\frac{-1376±64\sqrt{35}i}{4896} when ± is plus. Add -1376 to 64i\sqrt{35}.

$$x=\frac{-1376+64\sqrt{35}i}{4896}$$

Divide -1376+64i\sqrt{35} by 4896.

$$x=\frac{-43+2\sqrt{35}i}{153}$$

Now solve the equation x=\frac{-1376±64\sqrt{35}i}{4896} when ± is minus. Subtract 64i\sqrt{35} from -1376.

$$x=\frac{-64\sqrt{35}i-1376}{4896}$$

Divide -1376-64i\sqrt{35} by 4896.

$$x=\frac{-2\sqrt{35}i-43}{153}$$

The equation is now solved.

$$x=\frac{-43+2\sqrt{35}i}{153} x=\frac{-2\sqrt{35}i-43}{153}$$

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.

$$2448x^{2}+1376x+208=0$$

Subtract 208 from both sides of the equation.

$$2448x^{2}+1376x+208-208=-208$$

Subtracting 208 from itself leaves 0.

$$2448x^{2}+1376x=-208$$

Divide both sides by 2448.

$$\frac{2448x^{2}+1376x}{2448}=-\frac{208}{2448}$$

Dividing by 2448 undoes the multiplication by 2448.

$$x^{2}+\frac{1376}{2448}x=-\frac{208}{2448}$$

Reduce the fraction \frac{1376}{2448} to lowest terms by extracting and canceling out 16.

$$x^{2}+\frac{86}{153}x=-\frac{208}{2448}$$

Reduce the fraction \frac{-208}{2448} to lowest terms by extracting and canceling out 16.

$$x^{2}+\frac{86}{153}x=-\frac{13}{153}$$

Divide \frac{86}{153}, the coefficient of the x term, by 2 to get \frac{43}{153}. Then add the square of \frac{43}{153} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{86}{153}x+\left(\frac{43}{153}\right)^{2}=-\frac{13}{153}+\left(\frac{43}{153}\right)^{2}$$

Square \frac{43}{153} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{86}{153}x+\frac{1849}{23409}=-\frac{13}{153}+\frac{1849}{23409}$$

Add -\frac{13}{153} to \frac{1849}{23409} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{86}{153}x+\frac{1849}{23409}=-\frac{140}{23409}$$

Factor x^{2}+\frac{86}{153}x+\frac{1849}{23409}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{43}{153}\right)^{2}=-\frac{140}{23409}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{43}{153}\right)^{2}}=\sqrt{-\frac{140}{23409}}$$

Simplify.

$$x+\frac{43}{153}=\frac{2\sqrt{35}i}{153} x+\frac{43}{153}=-\frac{2\sqrt{35}i}{153}$$

Subtract \frac{43}{153} from both sides of the equation.

$$x=\frac{-43+2\sqrt{35}i}{153} x=\frac{-2\sqrt{35}i-43}{153}$$