Solve x+3/6divx=4x+6/9-4

$$\frac { x + 3 } { 6 } \div x = \frac { 4 x + 6 } { 9 } - 4$$

Solve for x

$$x = \frac{3 \sqrt{473} + 63}{16} \approx 8.015355595$$

$$x=\frac{63-3\sqrt{473}}{16}\approx -0.140355595$$

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9x, the least common multiple of x,9.

$$9\times \frac{x+3}{6}=x\left(4x+6\right)+9x\left(-4\right)$$

Express 9\times \frac{x+3}{6} as a single fraction.

$$\frac{9\left(x+3\right)}{6}=x\left(4x+6\right)+9x\left(-4\right)$$

Use the distributive property to multiply x by 4x+6.

$$\frac{9\left(x+3\right)}{6}=4x^{2}+6x+9x\left(-4\right)$$

Multiply 9 and -4 to get -36.

$$\frac{9\left(x+3\right)}{6}=4x^{2}+6x-36x$$

Combine 6x and -36x to get -30x.

$$\frac{9\left(x+3\right)}{6}=4x^{2}-30x$$

Use the distributive property to multiply 9 by x+3.

$$\frac{9x+27}{6}=4x^{2}-30x$$

Divide each term of 9x+27 by 6 to get \frac{3}{2}x+\frac{9}{2}.

$$\frac{3}{2}x+\frac{9}{2}=4x^{2}-30x$$

Subtract 4x^{2} from both sides.

$$\frac{3}{2}x+\frac{9}{2}-4x^{2}=-30x$$

Add 30x to both sides.

$$\frac{3}{2}x+\frac{9}{2}-4x^{2}+30x=0$$

Combine \frac{3}{2}x and 30x to get \frac{63}{2}x.

$$\frac{63}{2}x+\frac{9}{2}-4x^{2}=0$$

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.

$$-4x^{2}+\frac{63}{2}x+\frac{9}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, \frac{63}{2} for b, and \frac{9}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{63}{2}±\sqrt{\left(\frac{63}{2}\right)^{2}-4\left(-4\right)\times \frac{9}{2}}}{2\left(-4\right)}$$

Square \frac{63}{2} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{63}{2}±\sqrt{\frac{3969}{4}-4\left(-4\right)\times \frac{9}{2}}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$x=\frac{-\frac{63}{2}±\sqrt{\frac{3969}{4}+16\times \frac{9}{2}}}{2\left(-4\right)}$$

Multiply 16 times \frac{9}{2}.

$$x=\frac{-\frac{63}{2}±\sqrt{\frac{3969}{4}+72}}{2\left(-4\right)}$$

Add \frac{3969}{4} to 72.

$$x=\frac{-\frac{63}{2}±\sqrt{\frac{4257}{4}}}{2\left(-4\right)}$$

Take the square root of \frac{4257}{4}.

$$x=\frac{-\frac{63}{2}±\frac{3\sqrt{473}}{2}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$x=\frac{-\frac{63}{2}±\frac{3\sqrt{473}}{2}}{-8}$$

Now solve the equation x=\frac{-\frac{63}{2}±\frac{3\sqrt{473}}{2}}{-8} when ± is plus. Add -\frac{63}{2} to \frac{3\sqrt{473}}{2}.

$$x=\frac{3\sqrt{473}-63}{-8\times 2}$$

Divide \frac{-63+3\sqrt{473}}{2} by -8.

$$x=\frac{63-3\sqrt{473}}{16}$$

Now solve the equation x=\frac{-\frac{63}{2}±\frac{3\sqrt{473}}{2}}{-8} when ± is minus. Subtract \frac{3\sqrt{473}}{2} from -\frac{63}{2}.

$$x=\frac{-3\sqrt{473}-63}{-8\times 2}$$

Divide \frac{-63-3\sqrt{473}}{2} by -8.

$$x=\frac{3\sqrt{473}+63}{16}$$

The equation is now solved.

$$x=\frac{63-3\sqrt{473}}{16} x=\frac{3\sqrt{473}+63}{16}$$

Steps for Completing the Square

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 9x, the least common multiple of x,9.

$$9\times \frac{x+3}{6}=x\left(4x+6\right)+9x\left(-4\right)$$

Express 9\times \frac{x+3}{6} as a single fraction.

$$\frac{9\left(x+3\right)}{6}=x\left(4x+6\right)+9x\left(-4\right)$$

Use the distributive property to multiply x by 4x+6.

$$\frac{9\left(x+3\right)}{6}=4x^{2}+6x+9x\left(-4\right)$$

Multiply 9 and -4 to get -36.

$$\frac{9\left(x+3\right)}{6}=4x^{2}+6x-36x$$

Combine 6x and -36x to get -30x.

$$\frac{9\left(x+3\right)}{6}=4x^{2}-30x$$

Use the distributive property to multiply 9 by x+3.

$$\frac{9x+27}{6}=4x^{2}-30x$$

Divide each term of 9x+27 by 6 to get \frac{3}{2}x+\frac{9}{2}.

$$\frac{3}{2}x+\frac{9}{2}=4x^{2}-30x$$

Subtract 4x^{2} from both sides.

$$\frac{3}{2}x+\frac{9}{2}-4x^{2}=-30x$$

Add 30x to both sides.

$$\frac{3}{2}x+\frac{9}{2}-4x^{2}+30x=0$$

Combine \frac{3}{2}x and 30x to get \frac{63}{2}x.

$$\frac{63}{2}x+\frac{9}{2}-4x^{2}=0$$

Subtract \frac{9}{2} from both sides. Anything subtracted from zero gives its negation.

$$\frac{63}{2}x-4x^{2}=-\frac{9}{2}$$

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.

$$-4x^{2}+\frac{63}{2}x=-\frac{9}{2}$$

Divide both sides by -4.

$$\frac{-4x^{2}+\frac{63}{2}x}{-4}=-\frac{\frac{9}{2}}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$x^{2}+\frac{\frac{63}{2}}{-4}x=-\frac{\frac{9}{2}}{-4}$$

Divide \frac{63}{2} by -4.

$$x^{2}-\frac{63}{8}x=-\frac{\frac{9}{2}}{-4}$$

Divide -\frac{9}{2} by -4.

$$x^{2}-\frac{63}{8}x=\frac{9}{8}$$

Divide -\frac{63}{8}, the coefficient of the x term, by 2 to get -\frac{63}{16}. Then add the square of -\frac{63}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{63}{8}x+\left(-\frac{63}{16}\right)^{2}=\frac{9}{8}+\left(-\frac{63}{16}\right)^{2}$$

Square -\frac{63}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{63}{8}x+\frac{3969}{256}=\frac{9}{8}+\frac{3969}{256}$$

Add \frac{9}{8} to \frac{3969}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{63}{8}x+\frac{3969}{256}=\frac{4257}{256}$$

Factor x^{2}-\frac{63}{8}x+\frac{3969}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{63}{16}\right)^{2}=\frac{4257}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{63}{16}\right)^{2}}=\sqrt{\frac{4257}{256}}$$

Simplify.

$$x-\frac{63}{16}=\frac{3\sqrt{473}}{16} x-\frac{63}{16}=-\frac{3\sqrt{473}}{16}$$

Add \frac{63}{16} to both sides of the equation.

$$x=\frac{3\sqrt{473}+63}{16} x=\frac{63-3\sqrt{473}}{16}$$