Solve 5/4x^2+9/4(3x^2-4sqrt{6}x+8)=45/18

$$\frac { 5 } { 4 } x ^ { 2 } + \frac { 9 } { 4 } ( 3 x ^ { 2 } - 4 \sqrt { 6 } x + 8 ) = \frac { 45 } { 18 }$$

Solve for x (complex solution)

$$x=\frac{9\sqrt{6}+\sqrt{10}i}{16}\approx 1.37783798+0.197642354i$$

$$x=\frac{-\sqrt{10}i+9\sqrt{6}}{16}\approx 1.37783798-0.197642354i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{9}{4} by 3x^{2}-4\sqrt{6}x+8.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+18=\frac{45}{18}$$

Reduce the fraction \frac{45}{18} to lowest terms by extracting and canceling out 9.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+18=\frac{5}{2}$$

Subtract \frac{5}{2} from both sides.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+18-\frac{5}{2}=0$$

Subtract \frac{5}{2} from 18 to get \frac{31}{2}.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+\frac{31}{2}=0$$

Use the distributive property to multiply \frac{9}{4} by 3x^{2}-4\sqrt{6}x.

$$\frac{5}{4}x^{2}+\frac{27}{4}x^{2}-9\sqrt{6}x+\frac{31}{2}=0$$

Combine \frac{5}{4}x^{2} and \frac{27}{4}x^{2} to get 8x^{2}.

$$8x^{2}-9\sqrt{6}x+\frac{31}{2}=0$$

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.

$$8x^{2}+\left(-9\sqrt{6}\right)x+\frac{31}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -9\sqrt{6} for b, and \frac{31}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{\left(-9\sqrt{6}\right)^{2}-4\times 8\times \frac{31}{2}}}{2\times 8}$$

Square -9\sqrt{6}.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{486-4\times 8\times \frac{31}{2}}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{486-32\times \frac{31}{2}}}{2\times 8}$$

Multiply -32 times \frac{31}{2}.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{486-496}}{2\times 8}$$

Add 486 to -496.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{-10}}{2\times 8}$$

Take the square root of -10.

$$x=\frac{-\left(-9\sqrt{6}\right)±\sqrt{10}i}{2\times 8}$$

The opposite of -9\sqrt{6} is 9\sqrt{6}.

$$x=\frac{9\sqrt{6}±\sqrt{10}i}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{9\sqrt{6}±\sqrt{10}i}{16}$$

Now solve the equation x=\frac{9\sqrt{6}±\sqrt{10}i}{16} when ± is plus. Add 9\sqrt{6} to i\sqrt{10}.

$$x=\frac{9\sqrt{6}+\sqrt{10}i}{16}$$

Now solve the equation x=\frac{9\sqrt{6}±\sqrt{10}i}{16} when ± is minus. Subtract i\sqrt{10} from 9\sqrt{6}.

$$x=\frac{-\sqrt{10}i+9\sqrt{6}}{16}$$

The equation is now solved.

$$x=\frac{9\sqrt{6}+\sqrt{10}i}{16} x=\frac{-\sqrt{10}i+9\sqrt{6}}{16}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{9}{4} by 3x^{2}-4\sqrt{6}x+8.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+18=\frac{45}{18}$$

Reduce the fraction \frac{45}{18} to lowest terms by extracting and canceling out 9.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)+18=\frac{5}{2}$$

Subtract 18 from both sides.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)=\frac{5}{2}-18$$

Subtract 18 from \frac{5}{2} to get -\frac{31}{2}.

$$\frac{5}{4}x^{2}+\frac{9}{4}\left(3x^{2}-4\sqrt{6}x\right)=-\frac{31}{2}$$

Use the distributive property to multiply \frac{9}{4} by 3x^{2}-4\sqrt{6}x.

$$\frac{5}{4}x^{2}+\frac{27}{4}x^{2}-9\sqrt{6}x=-\frac{31}{2}$$

Combine \frac{5}{4}x^{2} and \frac{27}{4}x^{2} to get 8x^{2}.

$$8x^{2}-9\sqrt{6}x=-\frac{31}{2}$$

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.

$$8x^{2}+\left(-9\sqrt{6}\right)x=-\frac{31}{2}$$

Divide both sides by 8.

$$\frac{8x^{2}+\left(-9\sqrt{6}\right)x}{8}=-\frac{\frac{31}{2}}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x=-\frac{\frac{31}{2}}{8}$$

Divide -\frac{31}{2} by 8.

$$x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x=-\frac{31}{16}$$

Divide -\frac{9\sqrt{6}}{8}, the coefficient of the x term, by 2 to get -\frac{9\sqrt{6}}{16}. Then add the square of -\frac{9\sqrt{6}}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x+\left(-\frac{9\sqrt{6}}{16}\right)^{2}=-\frac{31}{16}+\left(-\frac{9\sqrt{6}}{16}\right)^{2}$$

Square -\frac{9\sqrt{6}}{16}.

$$x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x+\frac{243}{128}=-\frac{31}{16}+\frac{243}{128}$$

Add -\frac{31}{16} to \frac{243}{128} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x+\frac{243}{128}=-\frac{5}{128}$$

Factor x^{2}+\left(-\frac{9\sqrt{6}}{8}\right)x+\frac{243}{128}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9\sqrt{6}}{16}\right)^{2}=-\frac{5}{128}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9\sqrt{6}}{16}\right)^{2}}=\sqrt{-\frac{5}{128}}$$

Simplify.

$$x-\frac{9\sqrt{6}}{16}=\frac{\sqrt{10}i}{16} x-\frac{9\sqrt{6}}{16}=-\frac{\sqrt{10}i}{16}$$

Add \frac{9\sqrt{6}}{16} to both sides of the equation.

$$x=\frac{9\sqrt{6}+\sqrt{10}i}{16} x=\frac{-\sqrt{10}i+9\sqrt{6}}{16}$$