Solve x/2(x+34)=20*1775

$$\frac { x } { 2 } ( x + 34 ) = 20 \times 1775$$

Solve for x

$$x=-284$$

$$x=250$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$x\left(x+34\right)=40\times 1775$$

Use the distributive property to multiply x by x+34.

$$x^{2}+34x=40\times 1775$$

Multiply 40 and 1775 to get 71000.

$$x^{2}+34x=71000$$

Subtract 71000 from both sides.

$$x^{2}+34x-71000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 34 for b, and -71000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-34±\sqrt{34^{2}-4\left(-71000\right)}}{2}$$

Square 34.

$$x=\frac{-34±\sqrt{1156-4\left(-71000\right)}}{2}$$

Multiply -4 times -71000.

$$x=\frac{-34±\sqrt{1156+284000}}{2}$$

Add 1156 to 284000.

$$x=\frac{-34±\sqrt{285156}}{2}$$

Take the square root of 285156.

$$x=\frac{-34±534}{2}$$

Now solve the equation x=\frac{-34±534}{2} when ± is plus. Add -34 to 534.

$$x=\frac{500}{2}$$

Divide 500 by 2.

$$x=250$$

Now solve the equation x=\frac{-34±534}{2} when ± is minus. Subtract 534 from -34.

$$x=-\frac{568}{2}$$

Divide -568 by 2.

$$x=-284$$

The equation is now solved.

$$x=250 x=-284$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$x\left(x+34\right)=40\times 1775$$

Use the distributive property to multiply x by x+34.

$$x^{2}+34x=40\times 1775$$

Multiply 40 and 1775 to get 71000.

$$x^{2}+34x=71000$$

Divide 34, the coefficient of the x term, by 2 to get 17. Then add the square of 17 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+34x+17^{2}=71000+17^{2}$$

Square 17.

$$x^{2}+34x+289=71000+289$$

Add 71000 to 289.

$$x^{2}+34x+289=71289$$

Factor x^{2}+34x+289. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+17\right)^{2}=71289$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+17\right)^{2}}=\sqrt{71289}$$

Simplify.

$$x+17=267 x+17=-267$$

Subtract 17 from both sides of the equation.

$$x=250 x=-284$$