Solve 64left(5+x^2right)=(18-3x)2

$$64 \left( 5+ { x }^{ 2 } \right) = (18-3x)2$$

Solve for x (complex solution)

$$x=\frac{-3+\sqrt{18167}i}{64}\approx -0.046875+2.106015844i$$

$$x=\frac{-\sqrt{18167}i-3}{64}\approx -0.046875-2.106015844i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 64 by 5+x^{2}.

$$320+64x^{2}=\left(18-3x\right)\times 2$$

Use the distributive property to multiply 18-3x by 2.

$$320+64x^{2}=36-6x$$

Subtract 36 from both sides.

$$320+64x^{2}-36=-6x$$

Subtract 36 from 320 to get 284.

$$284+64x^{2}=-6x$$

Add 6x to both sides.

$$284+64x^{2}+6x=0$$

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.

$$64x^{2}+6x+284=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, 6 for b, and 284 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 64\times 284}}{2\times 64}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\times 64\times 284}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-6±\sqrt{36-256\times 284}}{2\times 64}$$

Multiply -256 times 284.

$$x=\frac{-6±\sqrt{36-72704}}{2\times 64}$$

Add 36 to -72704.

$$x=\frac{-6±\sqrt{-72668}}{2\times 64}$$

Take the square root of -72668.

$$x=\frac{-6±2\sqrt{18167}i}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{-6±2\sqrt{18167}i}{128}$$

Now solve the equation x=\frac{-6±2\sqrt{18167}i}{128} when ± is plus. Add -6 to 2i\sqrt{18167}.

$$x=\frac{-6+2\sqrt{18167}i}{128}$$

Divide -6+2i\sqrt{18167} by 128.

$$x=\frac{-3+\sqrt{18167}i}{64}$$

Now solve the equation x=\frac{-6±2\sqrt{18167}i}{128} when ± is minus. Subtract 2i\sqrt{18167} from -6.

$$x=\frac{-2\sqrt{18167}i-6}{128}$$

Divide -6-2i\sqrt{18167} by 128.

$$x=\frac{-\sqrt{18167}i-3}{64}$$

The equation is now solved.

$$x=\frac{-3+\sqrt{18167}i}{64} x=\frac{-\sqrt{18167}i-3}{64}$$

Steps for Completing the Square

Use the distributive property to multiply 64 by 5+x^{2}.

$$320+64x^{2}=\left(18-3x\right)\times 2$$

Use the distributive property to multiply 18-3x by 2.

$$320+64x^{2}=36-6x$$

Add 6x to both sides.

$$320+64x^{2}+6x=36$$

Subtract 320 from both sides.

$$64x^{2}+6x=36-320$$

Subtract 320 from 36 to get -284.

$$64x^{2}+6x=-284$$

Divide both sides by 64.

$$\frac{64x^{2}+6x}{64}=-\frac{284}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$x^{2}+\frac{6}{64}x=-\frac{284}{64}$$

Reduce the fraction \frac{6}{64} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{3}{32}x=-\frac{284}{64}$$

Reduce the fraction \frac{-284}{64} to lowest terms by extracting and canceling out 4.

$$x^{2}+\frac{3}{32}x=-\frac{71}{16}$$

Divide \frac{3}{32}, the coefficient of the x term, by 2 to get \frac{3}{64}. Then add the square of \frac{3}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{32}x+\left(\frac{3}{64}\right)^{2}=-\frac{71}{16}+\left(\frac{3}{64}\right)^{2}$$

Square \frac{3}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{32}x+\frac{9}{4096}=-\frac{71}{16}+\frac{9}{4096}$$

Add -\frac{71}{16} to \frac{9}{4096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{32}x+\frac{9}{4096}=-\frac{18167}{4096}$$

Factor x^{2}+\frac{3}{32}x+\frac{9}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{64}\right)^{2}=-\frac{18167}{4096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{64}\right)^{2}}=\sqrt{-\frac{18167}{4096}}$$

Simplify.

$$x+\frac{3}{64}=\frac{\sqrt{18167}i}{64} x+\frac{3}{64}=-\frac{\sqrt{18167}i}{64}$$

Subtract \frac{3}{64} from both sides of the equation.

$$x=\frac{-3+\sqrt{18167}i}{64} x=\frac{-\sqrt{18167}i-3}{64}$$