Solve 5x+2^15x^2-108=0

$$5 x + 2 ^ { 1 } 5 x ^ { 2 } - 108 = 0$$

Solve for x

$$x=\frac{\sqrt{4345}}{20}-\frac{1}{4}\approx 3.0458307$$

$$x=-\frac{\sqrt{4345}}{20}-\frac{1}{4}\approx -3.5458307$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 1 and get 2.

$$5x+2\times 5x^{2}-108=0$$

Multiply 2 and 5 to get 10.

$$5x+10x^{2}-108=0$$

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.

$$10x^{2}+5x-108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 10 for a, 5 for b, and -108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 10\left(-108\right)}}{2\times 10}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 10\left(-108\right)}}{2\times 10}$$

Multiply -4 times 10.

$$x=\frac{-5±\sqrt{25-40\left(-108\right)}}{2\times 10}$$

Multiply -40 times -108.

$$x=\frac{-5±\sqrt{25+4320}}{2\times 10}$$

Add 25 to 4320.

$$x=\frac{-5±\sqrt{4345}}{2\times 10}$$

Multiply 2 times 10.

$$x=\frac{-5±\sqrt{4345}}{20}$$

Now solve the equation x=\frac{-5±\sqrt{4345}}{20} when ± is plus. Add -5 to \sqrt{4345}.

$$x=\frac{\sqrt{4345}-5}{20}$$

Divide -5+\sqrt{4345} by 20.

$$x=\frac{\sqrt{4345}}{20}-\frac{1}{4}$$

Now solve the equation x=\frac{-5±\sqrt{4345}}{20} when ± is minus. Subtract \sqrt{4345} from -5.

$$x=\frac{-\sqrt{4345}-5}{20}$$

Divide -5-\sqrt{4345} by 20.

$$x=-\frac{\sqrt{4345}}{20}-\frac{1}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{4345}}{20}-\frac{1}{4} x=-\frac{\sqrt{4345}}{20}-\frac{1}{4}$$

Steps for Completing the Square

Calculate 2 to the power of 1 and get 2.

$$5x+2\times 5x^{2}-108=0$$

Multiply 2 and 5 to get 10.

$$5x+10x^{2}-108=0$$

Add 108 to both sides. Anything plus zero gives itself.

$$5x+10x^{2}=108$$

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.

$$10x^{2}+5x=108$$

Divide both sides by 10.

$$\frac{10x^{2}+5x}{10}=\frac{108}{10}$$

Dividing by 10 undoes the multiplication by 10.

$$x^{2}+\frac{5}{10}x=\frac{108}{10}$$

Reduce the fraction \frac{5}{10} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{1}{2}x=\frac{108}{10}$$

Reduce the fraction \frac{108}{10} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{2}x=\frac{54}{5}$$

Divide \frac{1}{2}, the coefficient of the x term, by 2 to get \frac{1}{4}. Then add the square of \frac{1}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{2}x+\left(\frac{1}{4}\right)^{2}=\frac{54}{5}+\left(\frac{1}{4}\right)^{2}$$

Square \frac{1}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{54}{5}+\frac{1}{16}$$

Add \frac{54}{5} to \frac{1}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=\frac{869}{80}$$

Factor x^{2}+\frac{1}{2}x+\frac{1}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{4}\right)^{2}=\frac{869}{80}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{\frac{869}{80}}$$

Simplify.

$$x+\frac{1}{4}=\frac{\sqrt{4345}}{20} x+\frac{1}{4}=-\frac{\sqrt{4345}}{20}$$

Subtract \frac{1}{4} from both sides of the equation.

$$x=\frac{\sqrt{4345}}{20}-\frac{1}{4} x=-\frac{\sqrt{4345}}{20}-\frac{1}{4}$$