Solve (26x+1.6)*x=29.6

$$(26x+1.6) \times x=29.6$$

Solve for x

$$x = \frac{\sqrt{4814} - 2}{65} \approx 1.036661462$$

$$x=\frac{-\sqrt{4814}-2}{65}\approx -1.098199924$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 26x+1.6 by x.

$$26x^{2}+1.6x=29.6$$

Subtract 29.6 from both sides.

$$26x^{2}+1.6x-29.6=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 26 for a, 1.6 for b, and -29.6 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1.6±\sqrt{1.6^{2}-4\times 26\left(-29.6\right)}}{2\times 26}$$

Square 1.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-1.6±\sqrt{2.56-4\times 26\left(-29.6\right)}}{2\times 26}$$

Multiply -4 times 26.

$$x=\frac{-1.6±\sqrt{2.56-104\left(-29.6\right)}}{2\times 26}$$

Multiply -104 times -29.6.

$$x=\frac{-1.6±\sqrt{2.56+3078.4}}{2\times 26}$$

Add 2.56 to 3078.4 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-1.6±\sqrt{3080.96}}{2\times 26}$$

Take the square root of 3080.96.

$$x=\frac{-1.6±\frac{4\sqrt{4814}}{5}}{2\times 26}$$

Multiply 2 times 26.

$$x=\frac{-1.6±\frac{4\sqrt{4814}}{5}}{52}$$

Now solve the equation x=\frac{-1.6±\frac{4\sqrt{4814}}{5}}{52} when ± is plus. Add -1.6 to \frac{4\sqrt{4814}}{5}.

$$x=\frac{4\sqrt{4814}-8}{5\times 52}$$

Divide \frac{-8+4\sqrt{4814}}{5} by 52.

$$x=\frac{\sqrt{4814}-2}{65}$$

Now solve the equation x=\frac{-1.6±\frac{4\sqrt{4814}}{5}}{52} when ± is minus. Subtract \frac{4\sqrt{4814}}{5} from -1.6.

$$x=\frac{-4\sqrt{4814}-8}{5\times 52}$$

Divide \frac{-8-4\sqrt{4814}}{5} by 52.

$$x=\frac{-\sqrt{4814}-2}{65}$$

The equation is now solved.

$$x=\frac{\sqrt{4814}-2}{65} x=\frac{-\sqrt{4814}-2}{65}$$

Steps for Completing the Square

Use the distributive property to multiply 26x+1.6 by x.

$$26x^{2}+1.6x=29.6$$

Divide both sides by 26.

$$\frac{26x^{2}+1.6x}{26}=\frac{29.6}{26}$$

Dividing by 26 undoes the multiplication by 26.

$$x^{2}+\frac{1.6}{26}x=\frac{29.6}{26}$$

Divide 1.6 by 26.

$$x^{2}+\frac{4}{65}x=\frac{29.6}{26}$$

Divide 29.6 by 26.

$$x^{2}+\frac{4}{65}x=\frac{74}{65}$$

Divide \frac{4}{65}, the coefficient of the x term, by 2 to get \frac{2}{65}. Then add the square of \frac{2}{65} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{4}{65}x+\frac{2}{65}^{2}=\frac{74}{65}+\frac{2}{65}^{2}$$

Square \frac{2}{65} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{4}{65}x+\frac{4}{4225}=\frac{74}{65}+\frac{4}{4225}$$

Add \frac{74}{65} to \frac{4}{4225} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{4}{65}x+\frac{4}{4225}=\frac{4814}{4225}$$

Factor x^{2}+\frac{4}{65}x+\frac{4}{4225}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{2}{65}\right)^{2}=\frac{4814}{4225}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{2}{65}\right)^{2}}=\sqrt{\frac{4814}{4225}}$$

Simplify.

$$x+\frac{2}{65}=\frac{\sqrt{4814}}{65} x+\frac{2}{65}=-\frac{\sqrt{4814}}{65}$$

Subtract \frac{2}{65} from both sides of the equation.

$$x=\frac{\sqrt{4814}-2}{65} x=\frac{-\sqrt{4814}-2}{65}$$