Solve x^2+0.6x+0.09-0.36x^2=0

$${ x }^{ 2 } +0.6x+0.09-0.36 { x }^{ 2 } =0$$

Solve for x

$$x=-0.75$$

$$x=-0.1875$$

Steps Using the Quadratic Formula

Combine x^{2} and -0.36x^{2} to get 0.64x^{2}.

$$0.64x^{2}+0.6x+0.09=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.64 for a, 0.6 for b, and 0.09 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-0.6±\sqrt{0.6^{2}-4\times 0.64\times 0.09}}{2\times 0.64}$$

Square 0.6 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-0.6±\sqrt{0.36-4\times 0.64\times 0.09}}{2\times 0.64}$$

Multiply -4 times 0.64.

$$x=\frac{-0.6±\sqrt{0.36-2.56\times 0.09}}{2\times 0.64}$$

Multiply -2.56 times 0.09 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.6±\sqrt{0.36-0.2304}}{2\times 0.64}$$

Add 0.36 to -0.2304 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-0.6±\sqrt{0.1296}}{2\times 0.64}$$

Take the square root of 0.1296.

$$x=\frac{-0.6±\frac{9}{25}}{2\times 0.64}$$

Multiply 2 times 0.64.

$$x=\frac{-0.6±\frac{9}{25}}{1.28}$$

Now solve the equation x=\frac{-0.6±\frac{9}{25}}{1.28} when ± is plus. Add -0.6 to \frac{9}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{6}{25}}{1.28}$$

Divide -\frac{6}{25} by 1.28 by multiplying -\frac{6}{25} by the reciprocal of 1.28.

$$x=-\frac{3}{16}$$

Now solve the equation x=\frac{-0.6±\frac{9}{25}}{1.28} when ± is minus. Subtract \frac{9}{25} from -0.6 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{\frac{24}{25}}{1.28}$$

Divide -\frac{24}{25} by 1.28 by multiplying -\frac{24}{25} by the reciprocal of 1.28.

$$x=-\frac{3}{4}$$

The equation is now solved.

$$x=-\frac{3}{16} x=-\frac{3}{4}$$

Steps for Completing the Square

Combine x^{2} and -0.36x^{2} to get 0.64x^{2}.

$$0.64x^{2}+0.6x+0.09=0$$

Subtract 0.09 from both sides. Anything subtracted from zero gives its negation.

$$0.64x^{2}+0.6x=-0.09$$

Divide both sides of the equation by 0.64, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{0.64x^{2}+0.6x}{0.64}=-\frac{0.09}{0.64}$$

Dividing by 0.64 undoes the multiplication by 0.64.

$$x^{2}+\frac{0.6}{0.64}x=-\frac{0.09}{0.64}$$

Divide 0.6 by 0.64 by multiplying 0.6 by the reciprocal of 0.64.

$$x^{2}+0.9375x=-\frac{0.09}{0.64}$$

Divide -0.09 by 0.64 by multiplying -0.09 by the reciprocal of 0.64.

$$x^{2}+0.9375x=-0.140625$$

Divide 0.9375, the coefficient of the x term, by 2 to get 0.46875. Then add the square of 0.46875 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+0.9375x+0.46875^{2}=-0.140625+0.46875^{2}$$

Square 0.46875 by squaring both the numerator and the denominator of the fraction.

$$x^{2}+0.9375x+0.2197265625=-0.140625+0.2197265625$$

Add -0.140625 to 0.2197265625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+0.9375x+0.2197265625=0.0791015625$$

Factor x^{2}+0.9375x+0.2197265625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+0.46875\right)^{2}=0.0791015625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+0.46875\right)^{2}}=\sqrt{0.0791015625}$$

Simplify.

$$x+0.46875=\frac{9}{32} x+0.46875=-\frac{9}{32}$$

Subtract 0.46875 from both sides of the equation.

$$x=-\frac{3}{16} x=-\frac{3}{4}$$