Solve 0.11x+0.011x^2=0.429

$$0.11x+0.011 { x }^{ 2 } =0.429$$

Solve for x

$$x=-13$$

$$x=3$$

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.

$$0.011x^{2}+0.11x=0.429$$

Subtract 0.429 from both sides of the equation.

$$0.011x^{2}+0.11x-0.429=0.429-0.429$$

Subtracting 0.429 from itself leaves 0.

$$0.011x^{2}+0.11x-0.429=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.011 for a, 0.11 for b, and -0.429 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.11±\sqrt{0.11^{2}-4\times 0.011\left(-0.429\right)}}{2\times 0.011}$$

Square 0.11 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.11±\sqrt{0.0121-4\times 0.011\left(-0.429\right)}}{2\times 0.011}$$

Multiply -4 times 0.011.

$$x=\frac{-0.11±\sqrt{0.0121-0.044\left(-0.429\right)}}{2\times 0.011}$$

Multiply -0.044 times -0.429 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.11±\sqrt{0.0121+0.018876}}{2\times 0.011}$$

Add 0.0121 to 0.018876 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.11±\sqrt{0.030976}}{2\times 0.011}$$

Take the square root of 0.030976.

$$x=\frac{-0.11±\frac{22}{125}}{2\times 0.011}$$

Multiply 2 times 0.011.

$$x=\frac{-0.11±\frac{22}{125}}{0.022}$$

Now solve the equation x=\frac{-0.11±\frac{22}{125}}{0.022} when ± is plus. Add -0.11 to \frac{22}{125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{\frac{33}{500}}{0.022}$$

Divide \frac{33}{500} by 0.022 by multiplying \frac{33}{500} by the reciprocal of 0.022.

$$x=3$$

Now solve the equation x=\frac{-0.11±\frac{22}{125}}{0.022} when ± is minus. Subtract \frac{22}{125} from -0.11 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{143}{500}}{0.022}$$

Divide -\frac{143}{500} by 0.022 by multiplying -\frac{143}{500} by the reciprocal of 0.022.

$$x=-13$$

The equation is now solved.

$$x=3 x=-13$$

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.

$$0.011x^{2}+0.11x=0.429$$

Divide both sides of the equation by 0.011, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.011x^{2}+0.11x}{0.011}=\frac{0.429}{0.011}$$

Dividing by 0.011 undoes the multiplication by 0.011.

$$x^{2}+\frac{0.11}{0.011}x=\frac{0.429}{0.011}$$

Divide 0.11 by 0.011 by multiplying 0.11 by the reciprocal of 0.011.

$$x^{2}+10x=\frac{0.429}{0.011}$$

Divide 0.429 by 0.011 by multiplying 0.429 by the reciprocal of 0.011.

$$x^{2}+10x=39$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10x+5^{2}=39+5^{2}$$

Square 5.

$$x^{2}+10x+25=39+25$$

Add 39 to 25.

$$x^{2}+10x+25=64$$

Factor x^{2}+10x+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5\right)^{2}=64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5\right)^{2}}=\sqrt{64}$$

Simplify.

$$x+5=8 x+5=-8$$

Subtract 5 from both sides of the equation.

$$x=3 x=-13$$