Solve 4a^2/1-a=1.898

$$\frac { 4 a ^ { 2 } } { 1 - a } = 1.898$$

Solve for a

$$a=\frac{\sqrt{8492601}-949}{4000}\approx 0.491301688$$

$$a=\frac{-\sqrt{8492601}-949}{4000}\approx -0.965801688$$

Steps Using the Quadratic Formula

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -a+1.

$$4a^{2}=1.898\left(-a+1\right)$$

Use the distributive property to multiply 1.898 by -a+1.

$$4a^{2}=-1.898a+1.898$$

Add 1.898a to both sides.

$$4a^{2}+1.898a=1.898$$

Subtract 1.898 from both sides.

$$4a^{2}+1.898a-1.898=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 1.898 for b, and -1.898 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-1.898±\sqrt{1.898^{2}-4\times 4\left(-1.898\right)}}{2\times 4}$$

Square 1.898 by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-1.898±\sqrt{3.602404-4\times 4\left(-1.898\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-1.898±\sqrt{3.602404-16\left(-1.898\right)}}{2\times 4}$$

Multiply -16 times -1.898.

$$a=\frac{-1.898±\sqrt{3.602404+30.368}}{2\times 4}$$

Add 3.602404 to 30.368 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a=\frac{-1.898±\sqrt{33.970404}}{2\times 4}$$

Take the square root of 33.970404.

$$a=\frac{-1.898±\frac{\sqrt{8492601}}{500}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-1.898±\frac{\sqrt{8492601}}{500}}{8}$$

Now solve the equation a=\frac{-1.898±\frac{\sqrt{8492601}}{500}}{8} when ± is plus. Add -1.898 to \frac{\sqrt{8492601}}{500}.

$$a=\frac{\sqrt{8492601}-949}{8\times 500}$$

Divide \frac{-949+\sqrt{8492601}}{500} by 8.

$$a=\frac{\sqrt{8492601}-949}{4000}$$

Now solve the equation a=\frac{-1.898±\frac{\sqrt{8492601}}{500}}{8} when ± is minus. Subtract \frac{\sqrt{8492601}}{500} from -1.898.

$$a=\frac{-\sqrt{8492601}-949}{8\times 500}$$

Divide \frac{-949-\sqrt{8492601}}{500} by 8.

$$a=\frac{-\sqrt{8492601}-949}{4000}$$

The equation is now solved.

$$a=\frac{\sqrt{8492601}-949}{4000} a=\frac{-\sqrt{8492601}-949}{4000}$$

Steps for Completing the Square

Variable a cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by -a+1.

$$4a^{2}=1.898\left(-a+1\right)$$

Use the distributive property to multiply 1.898 by -a+1.

$$4a^{2}=-1.898a+1.898$$

Add 1.898a to both sides.

$$4a^{2}+1.898a=1.898$$

Divide both sides by 4.

$$\frac{4a^{2}+1.898a}{4}=\frac{1.898}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}+\frac{1.898}{4}a=\frac{1.898}{4}$$

Divide 1.898 by 4.

$$a^{2}+0.4745a=\frac{1.898}{4}$$

Divide 1.898 by 4.

$$a^{2}+0.4745a=0.4745$$

Divide 0.4745, the coefficient of the x term, by 2 to get 0.23725. Then add the square of 0.23725 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+0.4745a+0.23725^{2}=0.4745+0.23725^{2}$$

Square 0.23725 by squaring both the numerator and the denominator of the fraction.

$$a^{2}+0.4745a+0.0562875625=0.4745+0.0562875625$$

Add 0.4745 to 0.0562875625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+0.4745a+0.0562875625=0.5307875625$$

Factor a^{2}+0.4745a+0.0562875625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+0.23725\right)^{2}=0.5307875625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+0.23725\right)^{2}}=\sqrt{0.5307875625}$$

Simplify.

$$a+0.23725=\frac{\sqrt{8492601}}{4000} a+0.23725=-\frac{\sqrt{8492601}}{4000}$$

Subtract 0.23725 from both sides of the equation.

$$a=\frac{\sqrt{8492601}-949}{4000} a=\frac{-\sqrt{8492601}-949}{4000}$$