Solve 30705=x/2(2(17)+(x-1)(3))

$$30705= \frac{ x }{ 2 } (2(17)+(x-1)(3))$$

Solve for x

$$x = -\frac{445}{3} = -148\frac{1}{3} \approx -148.333333333$$

$$x=138$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$61410=x\left(2\times 17+\left(x-1\right)\times 3\right)$$

Multiply 2 and 17 to get 34.

$$61410=x\left(34+\left(x-1\right)\times 3\right)$$

Use the distributive property to multiply x-1 by 3.

$$61410=x\left(34+3x-3\right)$$

Subtract 3 from 34 to get 31.

$$61410=x\left(31+3x\right)$$

Use the distributive property to multiply x by 31+3x.

$$61410=31x+3x^{2}$$

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

$$31x+3x^{2}=61410$$

Subtract 61410 from both sides.

$$31x+3x^{2}-61410=0$$

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.

$$3x^{2}+31x-61410=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 31 for b, and -61410 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-31±\sqrt{31^{2}-4\times 3\left(-61410\right)}}{2\times 3}$$

Square 31.

$$x=\frac{-31±\sqrt{961-4\times 3\left(-61410\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-31±\sqrt{961-12\left(-61410\right)}}{2\times 3}$$

Multiply -12 times -61410.

$$x=\frac{-31±\sqrt{961+736920}}{2\times 3}$$

Add 961 to 736920.

$$x=\frac{-31±\sqrt{737881}}{2\times 3}$$

Take the square root of 737881.

$$x=\frac{-31±859}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-31±859}{6}$$

Now solve the equation x=\frac{-31±859}{6} when ± is plus. Add -31 to 859.

$$x=\frac{828}{6}$$

Divide 828 by 6.

$$x=138$$

Now solve the equation x=\frac{-31±859}{6} when ± is minus. Subtract 859 from -31.

$$x=-\frac{890}{6}$$

Reduce the fraction \frac{-890}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{445}{3}$$

The equation is now solved.

$$x=138 x=-\frac{445}{3}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$61410=x\left(2\times 17+\left(x-1\right)\times 3\right)$$

Multiply 2 and 17 to get 34.

$$61410=x\left(34+\left(x-1\right)\times 3\right)$$

Use the distributive property to multiply x-1 by 3.

$$61410=x\left(34+3x-3\right)$$

Subtract 3 from 34 to get 31.

$$61410=x\left(31+3x\right)$$

Use the distributive property to multiply x by 31+3x.

$$61410=31x+3x^{2}$$

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

$$31x+3x^{2}=61410$$

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.

$$3x^{2}+31x=61410$$

Divide both sides by 3.

$$\frac{3x^{2}+31x}{3}=\frac{61410}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{31}{3}x=\frac{61410}{3}$$

Divide 61410 by 3.

$$x^{2}+\frac{31}{3}x=20470$$

Divide \frac{31}{3}, the coefficient of the x term, by 2 to get \frac{31}{6}. Then add the square of \frac{31}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{31}{3}x+\left(\frac{31}{6}\right)^{2}=20470+\left(\frac{31}{6}\right)^{2}$$

Square \frac{31}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{31}{3}x+\frac{961}{36}=20470+\frac{961}{36}$$

Add 20470 to \frac{961}{36}.

$$x^{2}+\frac{31}{3}x+\frac{961}{36}=\frac{737881}{36}$$

Factor x^{2}+\frac{31}{3}x+\frac{961}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{31}{6}\right)^{2}=\frac{737881}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{31}{6}\right)^{2}}=\sqrt{\frac{737881}{36}}$$

Simplify.

$$x+\frac{31}{6}=\frac{859}{6} x+\frac{31}{6}=-\frac{859}{6}$$

Subtract \frac{31}{6} from both sides of the equation.

$$x=138 x=-\frac{445}{3}$$