Solve x-(258x)-(400x*x)-(10*x)=45

$$x - ( 258 x ) - ( 400 x \times x ) - ( 10 * x ) = 45$$

Solve for x (complex solution)

$$x=\frac{-3\sqrt{79}i-267}{800}\approx -0.33375-0.033330729i$$

$$x=\frac{-267+3\sqrt{79}i}{800}\approx -0.33375+0.033330729i$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

$$x-258x-400x^{2}-10x=45$$

Combine x and -258x to get -257x.

$$-257x-400x^{2}-10x=45$$

Combine -257x and -10x to get -267x.

$$-267x-400x^{2}=45$$

Subtract 45 from both sides.

$$-267x-400x^{2}-45=0$$

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.

$$-400x^{2}-267x-45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -400 for a, -267 for b, and -45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-267\right)±\sqrt{\left(-267\right)^{2}-4\left(-400\right)\left(-45\right)}}{2\left(-400\right)}$$

Square -267.

$$x=\frac{-\left(-267\right)±\sqrt{71289-4\left(-400\right)\left(-45\right)}}{2\left(-400\right)}$$

Multiply -4 times -400.

$$x=\frac{-\left(-267\right)±\sqrt{71289+1600\left(-45\right)}}{2\left(-400\right)}$$

Multiply 1600 times -45.

$$x=\frac{-\left(-267\right)±\sqrt{71289-72000}}{2\left(-400\right)}$$

Add 71289 to -72000.

$$x=\frac{-\left(-267\right)±\sqrt{-711}}{2\left(-400\right)}$$

Take the square root of -711.

$$x=\frac{-\left(-267\right)±3\sqrt{79}i}{2\left(-400\right)}$$

The opposite of -267 is 267.

$$x=\frac{267±3\sqrt{79}i}{2\left(-400\right)}$$

Multiply 2 times -400.

$$x=\frac{267±3\sqrt{79}i}{-800}$$

Now solve the equation x=\frac{267±3\sqrt{79}i}{-800} when ± is plus. Add 267 to 3i\sqrt{79}.

$$x=\frac{267+3\sqrt{79}i}{-800}$$

Divide 267+3i\sqrt{79} by -800.

$$x=\frac{-3\sqrt{79}i-267}{800}$$

Now solve the equation x=\frac{267±3\sqrt{79}i}{-800} when ± is minus. Subtract 3i\sqrt{79} from 267.

$$x=\frac{-3\sqrt{79}i+267}{-800}$$

Divide 267-3i\sqrt{79} by -800.

$$x=\frac{-267+3\sqrt{79}i}{800}$$

The equation is now solved.

$$x=\frac{-3\sqrt{79}i-267}{800} x=\frac{-267+3\sqrt{79}i}{800}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

$$x-258x-400x^{2}-10x=45$$

Combine x and -258x to get -257x.

$$-257x-400x^{2}-10x=45$$

Combine -257x and -10x to get -267x.

$$-267x-400x^{2}=45$$

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.

$$-400x^{2}-267x=45$$

Divide both sides by -400.

$$\frac{-400x^{2}-267x}{-400}=\frac{45}{-400}$$

Dividing by -400 undoes the multiplication by -400.

$$x^{2}+\left(-\frac{267}{-400}\right)x=\frac{45}{-400}$$

Divide -267 by -400.

$$x^{2}+\frac{267}{400}x=\frac{45}{-400}$$

Reduce the fraction \frac{45}{-400} to lowest terms by extracting and canceling out 5.

$$x^{2}+\frac{267}{400}x=-\frac{9}{80}$$

Divide \frac{267}{400}, the coefficient of the x term, by 2 to get \frac{267}{800}. Then add the square of \frac{267}{800} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{267}{400}x+\left(\frac{267}{800}\right)^{2}=-\frac{9}{80}+\left(\frac{267}{800}\right)^{2}$$

Square \frac{267}{800} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{267}{400}x+\frac{71289}{640000}=-\frac{9}{80}+\frac{71289}{640000}$$

Add -\frac{9}{80} to \frac{71289}{640000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{267}{400}x+\frac{71289}{640000}=-\frac{711}{640000}$$

Factor x^{2}+\frac{267}{400}x+\frac{71289}{640000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{267}{800}\right)^{2}=-\frac{711}{640000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{267}{800}\right)^{2}}=\sqrt{-\frac{711}{640000}}$$

Simplify.

$$x+\frac{267}{800}=\frac{3\sqrt{79}i}{800} x+\frac{267}{800}=-\frac{3\sqrt{79}i}{800}$$

Subtract \frac{267}{800} from both sides of the equation.

$$x=\frac{-267+3\sqrt{79}i}{800} x=\frac{-3\sqrt{79}i-267}{800}$$