Solve 4a^2/1-a=628.5

$$\frac { 4 a ^ { 2 } } { 1 - a } = 628.5$$

Solve for a

$$a=\frac{\sqrt{1620273}-1257}{16}\approx 0.993715384$$

$$a=\frac{-\sqrt{1620273}-1257}{16}\approx -158.118715384$$

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.

$$4a^{2}=628.5\left(-a+1\right)$$

Use the distributive property to multiply 628.5 by -a+1.

$$4a^{2}=-628.5a+628.5$$

Add 628.5a to both sides.

$$4a^{2}+628.5a=628.5$$

Subtract 628.5 from both sides.

$$4a^{2}+628.5a-628.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 628.5 for b, and -628.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-628.5±\sqrt{628.5^{2}-4\times 4\left(-628.5\right)}}{2\times 4}$$

Square 628.5 by squaring both the numerator and the denominator of the fraction.

$$a=\frac{-628.5±\sqrt{395012.25-4\times 4\left(-628.5\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-628.5±\sqrt{395012.25-16\left(-628.5\right)}}{2\times 4}$$

Multiply -16 times -628.5.

$$a=\frac{-628.5±\sqrt{395012.25+10056}}{2\times 4}$$

Add 395012.25 to 10056.

$$a=\frac{-628.5±\sqrt{405068.25}}{2\times 4}$$

Take the square root of 405068.25.

$$a=\frac{-628.5±\frac{\sqrt{1620273}}{2}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-628.5±\frac{\sqrt{1620273}}{2}}{8}$$

Now solve the equation a=\frac{-628.5±\frac{\sqrt{1620273}}{2}}{8} when ± is plus. Add -628.5 to \frac{\sqrt{1620273}}{2}.

$$a=\frac{\sqrt{1620273}-1257}{2\times 8}$$

Divide \frac{-1257+\sqrt{1620273}}{2} by 8.

$$a=\frac{\sqrt{1620273}-1257}{16}$$

Now solve the equation a=\frac{-628.5±\frac{\sqrt{1620273}}{2}}{8} when ± is minus. Subtract \frac{\sqrt{1620273}}{2} from -628.5.

$$a=\frac{-\sqrt{1620273}-1257}{2\times 8}$$

Divide \frac{-1257-\sqrt{1620273}}{2} by 8.

$$a=\frac{-\sqrt{1620273}-1257}{16}$$

The equation is now solved.

$$a=\frac{\sqrt{1620273}-1257}{16} a=\frac{-\sqrt{1620273}-1257}{16}$$

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.

$$4a^{2}=628.5\left(-a+1\right)$$

Use the distributive property to multiply 628.5 by -a+1.

$$4a^{2}=-628.5a+628.5$$

Add 628.5a to both sides.

$$4a^{2}+628.5a=628.5$$

Divide both sides by 4.

$$\frac{4a^{2}+628.5a}{4}=\frac{628.5}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$a^{2}+\frac{628.5}{4}a=\frac{628.5}{4}$$

Divide 628.5 by 4.

$$a^{2}+157.125a=\frac{628.5}{4}$$

Divide 628.5 by 4.

$$a^{2}+157.125a=157.125$$

Divide 157.125, the coefficient of the x term, by 2 to get 78.5625. Then add the square of 78.5625 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}+157.125a+78.5625^{2}=157.125+78.5625^{2}$$

Square 78.5625 by squaring both the numerator and the denominator of the fraction.

$$a^{2}+157.125a+6172.06640625=157.125+6172.06640625$$

Add 157.125 to 6172.06640625 by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}+157.125a+6172.06640625=6329.19140625$$

Factor a^{2}+157.125a+6172.06640625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a+78.5625\right)^{2}=6329.19140625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+78.5625\right)^{2}}=\sqrt{6329.19140625}$$

Simplify.

$$a+78.5625=\frac{\sqrt{1620273}}{16} a+78.5625=-\frac{\sqrt{1620273}}{16}$$

Subtract 78.5625 from both sides of the equation.

$$a=\frac{\sqrt{1620273}-1257}{16} a=\frac{-\sqrt{1620273}-1257}{16}$$