Solve (x)=-1/2x^2+21x^2-7x+100

$$( x ) = - \frac { 1 } { 2 } x ^ { 2 } + 21 x ^ { 2 } - 7 x + 100$$

Solve for x (complex solution)

$$x=\frac{8+6\sqrt{226}i}{41}\approx 0.195121951+2.199994592i$$

$$x=\frac{-6\sqrt{226}i+8}{41}\approx 0.195121951-2.199994592i$$

Steps Using the Quadratic Formula

Combine -\frac{1}{2}x^{2} and 21x^{2} to get \frac{41}{2}x^{2}.

$$x=\frac{41}{2}x^{2}-7x+100$$

Subtract \frac{41}{2}x^{2} from both sides.

$$x-\frac{41}{2}x^{2}=-7x+100$$

Add 7x to both sides.

$$x-\frac{41}{2}x^{2}+7x=100$$

Combine x and 7x to get 8x.

$$8x-\frac{41}{2}x^{2}=100$$

Subtract 100 from both sides.

$$8x-\frac{41}{2}x^{2}-100=0$$

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.

$$-\frac{41}{2}x^{2}+8x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{41}{2} for a, 8 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-\frac{41}{2}\right)\left(-100\right)}}{2\left(-\frac{41}{2}\right)}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\left(-\frac{41}{2}\right)\left(-100\right)}}{2\left(-\frac{41}{2}\right)}$$

Multiply -4 times -\frac{41}{2}.

$$x=\frac{-8±\sqrt{64+82\left(-100\right)}}{2\left(-\frac{41}{2}\right)}$$

Multiply 82 times -100.

$$x=\frac{-8±\sqrt{64-8200}}{2\left(-\frac{41}{2}\right)}$$

Add 64 to -8200.

$$x=\frac{-8±\sqrt{-8136}}{2\left(-\frac{41}{2}\right)}$$

Take the square root of -8136.

$$x=\frac{-8±6\sqrt{226}i}{2\left(-\frac{41}{2}\right)}$$

Multiply 2 times -\frac{41}{2}.

$$x=\frac{-8±6\sqrt{226}i}{-41}$$

Now solve the equation x=\frac{-8±6\sqrt{226}i}{-41} when ± is plus. Add -8 to 6i\sqrt{226}.

$$x=\frac{-8+6\sqrt{226}i}{-41}$$

Divide -8+6i\sqrt{226} by -41.

$$x=\frac{-6\sqrt{226}i+8}{41}$$

Now solve the equation x=\frac{-8±6\sqrt{226}i}{-41} when ± is minus. Subtract 6i\sqrt{226} from -8.

$$x=\frac{-6\sqrt{226}i-8}{-41}$$

Divide -8-6i\sqrt{226} by -41.

$$x=\frac{8+6\sqrt{226}i}{41}$$

The equation is now solved.

$$x=\frac{-6\sqrt{226}i+8}{41} x=\frac{8+6\sqrt{226}i}{41}$$

Steps for Completing the Square

Combine -\frac{1}{2}x^{2} and 21x^{2} to get \frac{41}{2}x^{2}.

$$x=\frac{41}{2}x^{2}-7x+100$$

Subtract \frac{41}{2}x^{2} from both sides.

$$x-\frac{41}{2}x^{2}=-7x+100$$

Add 7x to both sides.

$$x-\frac{41}{2}x^{2}+7x=100$$

Combine x and 7x to get 8x.

$$8x-\frac{41}{2}x^{2}=100$$

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.

$$-\frac{41}{2}x^{2}+8x=100$$

Divide both sides of the equation by -\frac{41}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{41}{2}x^{2}+8x}{-\frac{41}{2}}=\frac{100}{-\frac{41}{2}}$$

Dividing by -\frac{41}{2} undoes the multiplication by -\frac{41}{2}.

$$x^{2}+\frac{8}{-\frac{41}{2}}x=\frac{100}{-\frac{41}{2}}$$

Divide 8 by -\frac{41}{2} by multiplying 8 by the reciprocal of -\frac{41}{2}.

$$x^{2}-\frac{16}{41}x=\frac{100}{-\frac{41}{2}}$$

Divide 100 by -\frac{41}{2} by multiplying 100 by the reciprocal of -\frac{41}{2}.

$$x^{2}-\frac{16}{41}x=-\frac{200}{41}$$

Divide -\frac{16}{41}, the coefficient of the x term, by 2 to get -\frac{8}{41}. Then add the square of -\frac{8}{41} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{16}{41}x+\left(-\frac{8}{41}\right)^{2}=-\frac{200}{41}+\left(-\frac{8}{41}\right)^{2}$$

Square -\frac{8}{41} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{16}{41}x+\frac{64}{1681}=-\frac{200}{41}+\frac{64}{1681}$$

Add -\frac{200}{41} to \frac{64}{1681} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{16}{41}x+\frac{64}{1681}=-\frac{8136}{1681}$$

Factor x^{2}-\frac{16}{41}x+\frac{64}{1681}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{8}{41}\right)^{2}=-\frac{8136}{1681}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{8}{41}\right)^{2}}=\sqrt{-\frac{8136}{1681}}$$

Simplify.

$$x-\frac{8}{41}=\frac{6\sqrt{226}i}{41} x-\frac{8}{41}=-\frac{6\sqrt{226}i}{41}$$

Add \frac{8}{41} to both sides of the equation.

$$x=\frac{8+6\sqrt{226}i}{41} x=\frac{-6\sqrt{226}i+8}{41}$$