Solve 2/x^2-9-7x/x-3+3=1000000

$$\frac{ 2 }{ { x }^{ 2 } -9 } - \frac{ 7x }{ x-3 } +3=1000000$$

Solve for x

$$x = \frac{\sqrt{36000044000041} - 21}{2000008} \approx 2.999979333$$

$$x=\frac{-\sqrt{36000044000041}-21}{2000008}\approx -3.000000333$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-9,x-3.

$$2-\left(x+3\right)\times 7x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 7.

$$2-\left(7x+21\right)x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 7x+21 by x.

$$2-\left(7x^{2}+21x\right)+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

To find the opposite of 7x^{2}+21x, find the opposite of each term.

$$2-7x^{2}-21x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$2-7x^{2}-21x+\left(x^{2}-9\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x^{2}-9 by 3.

$$2-7x^{2}-21x+3x^{2}-27=1000000\left(x-3\right)\left(x+3\right)$$

Combine -7x^{2} and 3x^{2} to get -4x^{2}.

$$2-4x^{2}-21x-27=1000000\left(x-3\right)\left(x+3\right)$$

Subtract 27 from 2 to get -25.

$$-25-4x^{2}-21x=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 1000000 by x-3.

$$-25-4x^{2}-21x=\left(1000000x-3000000\right)\left(x+3\right)$$

Use the distributive property to multiply 1000000x-3000000 by x+3 and combine like terms.

$$-25-4x^{2}-21x=1000000x^{2}-9000000$$

Subtract 1000000x^{2} from both sides.

$$-25-4x^{2}-21x-1000000x^{2}=-9000000$$

Combine -4x^{2} and -1000000x^{2} to get -1000004x^{2}.

$$-25-1000004x^{2}-21x=-9000000$$

Add 9000000 to both sides.

$$-25-1000004x^{2}-21x+9000000=0$$

Add -25 and 9000000 to get 8999975.

$$8999975-1000004x^{2}-21x=0$$

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.

$$-1000004x^{2}-21x+8999975=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1000004 for a, -21 for b, and 8999975 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-1000004\right)\times 8999975}}{2\left(-1000004\right)}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\left(-1000004\right)\times 8999975}}{2\left(-1000004\right)}$$

Multiply -4 times -1000004.

$$x=\frac{-\left(-21\right)±\sqrt{441+4000016\times 8999975}}{2\left(-1000004\right)}$$

Multiply 4000016 times 8999975.

$$x=\frac{-\left(-21\right)±\sqrt{441+36000043999600}}{2\left(-1000004\right)}$$

Add 441 to 36000043999600.

$$x=\frac{-\left(-21\right)±\sqrt{36000044000041}}{2\left(-1000004\right)}$$

The opposite of -21 is 21.

$$x=\frac{21±\sqrt{36000044000041}}{2\left(-1000004\right)}$$

Multiply 2 times -1000004.

$$x=\frac{21±\sqrt{36000044000041}}{-2000008}$$

Now solve the equation x=\frac{21±\sqrt{36000044000041}}{-2000008} when ± is plus. Add 21 to \sqrt{36000044000041}.

$$x=\frac{\sqrt{36000044000041}+21}{-2000008}$$

Divide 21+\sqrt{36000044000041} by -2000008.

$$x=\frac{-\sqrt{36000044000041}-21}{2000008}$$

Now solve the equation x=\frac{21±\sqrt{36000044000041}}{-2000008} when ± is minus. Subtract \sqrt{36000044000041} from 21.

$$x=\frac{21-\sqrt{36000044000041}}{-2000008}$$

Divide 21-\sqrt{36000044000041} by -2000008.

$$x=\frac{\sqrt{36000044000041}-21}{2000008}$$

The equation is now solved.

$$x=\frac{-\sqrt{36000044000041}-21}{2000008} x=\frac{\sqrt{36000044000041}-21}{2000008}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-9,x-3.

$$2-\left(x+3\right)\times 7x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x+3 by 7.

$$2-\left(7x+21\right)x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 7x+21 by x.

$$2-\left(7x^{2}+21x\right)+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

To find the opposite of 7x^{2}+21x, find the opposite of each term.

$$2-7x^{2}-21x+\left(x-3\right)\left(x+3\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x-3 by x+3 and combine like terms.

$$2-7x^{2}-21x+\left(x^{2}-9\right)\times 3=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply x^{2}-9 by 3.

$$2-7x^{2}-21x+3x^{2}-27=1000000\left(x-3\right)\left(x+3\right)$$

Combine -7x^{2} and 3x^{2} to get -4x^{2}.

$$2-4x^{2}-21x-27=1000000\left(x-3\right)\left(x+3\right)$$

Subtract 27 from 2 to get -25.

$$-25-4x^{2}-21x=1000000\left(x-3\right)\left(x+3\right)$$

Use the distributive property to multiply 1000000 by x-3.

$$-25-4x^{2}-21x=\left(1000000x-3000000\right)\left(x+3\right)$$

Use the distributive property to multiply 1000000x-3000000 by x+3 and combine like terms.

$$-25-4x^{2}-21x=1000000x^{2}-9000000$$

Subtract 1000000x^{2} from both sides.

$$-25-4x^{2}-21x-1000000x^{2}=-9000000$$

Combine -4x^{2} and -1000000x^{2} to get -1000004x^{2}.

$$-25-1000004x^{2}-21x=-9000000$$

Add 25 to both sides.

$$-1000004x^{2}-21x=-9000000+25$$

Add -9000000 and 25 to get -8999975.

$$-1000004x^{2}-21x=-8999975$$

Divide both sides by -1000004.

$$\frac{-1000004x^{2}-21x}{-1000004}=-\frac{8999975}{-1000004}$$

Dividing by -1000004 undoes the multiplication by -1000004.

$$x^{2}+\left(-\frac{21}{-1000004}\right)x=-\frac{8999975}{-1000004}$$

Divide -21 by -1000004.

$$x^{2}+\frac{21}{1000004}x=-\frac{8999975}{-1000004}$$

Divide -8999975 by -1000004.

$$x^{2}+\frac{21}{1000004}x=\frac{8999975}{1000004}$$

Divide \frac{21}{1000004}, the coefficient of the x term, by 2 to get \frac{21}{2000008}. Then add the square of \frac{21}{2000008} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{21}{1000004}x+\left(\frac{21}{2000008}\right)^{2}=\frac{8999975}{1000004}+\left(\frac{21}{2000008}\right)^{2}$$

Square \frac{21}{2000008} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{21}{1000004}x+\frac{441}{4000032000064}=\frac{8999975}{1000004}+\frac{441}{4000032000064}$$

Add \frac{8999975}{1000004} to \frac{441}{4000032000064} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{21}{1000004}x+\frac{441}{4000032000064}=\frac{36000044000041}{4000032000064}$$

Factor x^{2}+\frac{21}{1000004}x+\frac{441}{4000032000064}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{21}{2000008}\right)^{2}=\frac{36000044000041}{4000032000064}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{2000008}\right)^{2}}=\sqrt{\frac{36000044000041}{4000032000064}}$$

Simplify.

$$x+\frac{21}{2000008}=\frac{\sqrt{36000044000041}}{2000008} x+\frac{21}{2000008}=-\frac{\sqrt{36000044000041}}{2000008}$$

Subtract \frac{21}{2000008} from both sides of the equation.

$$x=\frac{\sqrt{36000044000041}-21}{2000008} x=\frac{-\sqrt{36000044000041}-21}{2000008}$$