Solve 5310=x(354-3x)/2

$$5310= \frac{ x(354-3x) }{ 2 }$$

Solve for x (complex solution)

$$x=59+\sqrt{59}i\approx 59+7.681145748i$$

$$x=-\sqrt{59}i+59\approx 59-7.681145748i$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$5310\times 2=x\left(354-3x\right)$$

Multiply 5310 and 2 to get 10620.

$$10620=x\left(354-3x\right)$$

Use the distributive property to multiply x by 354-3x.

$$10620=354x-3x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$354x-3x^{2}=10620$$

Subtract 10620 from both sides.

$$354x-3x^{2}-10620=0$$

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.

$$-3x^{2}+354x-10620=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 354 for b, and -10620 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-354±\sqrt{354^{2}-4\left(-3\right)\left(-10620\right)}}{2\left(-3\right)}$$

Square 354.

$$x=\frac{-354±\sqrt{125316-4\left(-3\right)\left(-10620\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-354±\sqrt{125316+12\left(-10620\right)}}{2\left(-3\right)}$$

Multiply 12 times -10620.

$$x=\frac{-354±\sqrt{125316-127440}}{2\left(-3\right)}$$

Add 125316 to -127440.

$$x=\frac{-354±\sqrt{-2124}}{2\left(-3\right)}$$

Take the square root of -2124.

$$x=\frac{-354±6\sqrt{59}i}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-354±6\sqrt{59}i}{-6}$$

Now solve the equation x=\frac{-354±6\sqrt{59}i}{-6} when ± is plus. Add -354 to 6i\sqrt{59}.

$$x=\frac{-354+6\sqrt{59}i}{-6}$$

Divide -354+6i\sqrt{59} by -6.

$$x=-\sqrt{59}i+59$$

Now solve the equation x=\frac{-354±6\sqrt{59}i}{-6} when ± is minus. Subtract 6i\sqrt{59} from -354.

$$x=\frac{-6\sqrt{59}i-354}{-6}$$

Divide -354-6i\sqrt{59} by -6.

$$x=59+\sqrt{59}i$$

The equation is now solved.

$$x=-\sqrt{59}i+59 x=59+\sqrt{59}i$$

Steps for Completing the Square

Multiply both sides by 2.

$$5310\times 2=x\left(354-3x\right)$$

Multiply 5310 and 2 to get 10620.

$$10620=x\left(354-3x\right)$$

Use the distributive property to multiply x by 354-3x.

$$10620=354x-3x^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$354x-3x^{2}=10620$$

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.

$$-3x^{2}+354x=10620$$

Divide both sides by -3.

$$\frac{-3x^{2}+354x}{-3}=\frac{10620}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{354}{-3}x=\frac{10620}{-3}$$

Divide 354 by -3.

$$x^{2}-118x=\frac{10620}{-3}$$

Divide 10620 by -3.

$$x^{2}-118x=-3540$$

Divide -118, the coefficient of the x term, by 2 to get -59. Then add the square of -59 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-118x+\left(-59\right)^{2}=-3540+\left(-59\right)^{2}$$

Square -59.

$$x^{2}-118x+3481=-3540+3481$$

Add -3540 to 3481.

$$x^{2}-118x+3481=-59$$

Factor x^{2}-118x+3481. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-59\right)^{2}=-59$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-59\right)^{2}}=\sqrt{-59}$$

Simplify.

$$x-59=\sqrt{59}i x-59=-\sqrt{59}i$$

Add 59 to both sides of the equation.

$$x=59+\sqrt{59}i x=-\sqrt{59}i+59$$