Solve -32x^2/55^2+x+200=100

$$\frac{ -32 { x }^{ 2 } }{ { 55 }^{ 2 } } +x+200=100$$

Solve for x

$$x = \frac{275 \sqrt{633} + 3025}{64} \approx 155.372813968$$

$$x=\frac{3025-275\sqrt{633}}{64}\approx -60.841563968$$

Steps Using the Quadratic Formula

Calculate 55 to the power of 2 and get 3025.

$$\frac{-32x^{2}}{3025}+x+200=100$$

Subtract 100 from both sides.

$$\frac{-32x^{2}}{3025}+x+200-100=0$$

Subtract 100 from 200 to get 100.

$$\frac{-32x^{2}}{3025}+x+100=0$$

Multiply both sides of the equation by 3025.

$$-32x^{2}+3025x+302500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -32 for a, 3025 for b, and 302500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3025±\sqrt{3025^{2}-4\left(-32\right)\times 302500}}{2\left(-32\right)}$$

Square 3025.

$$x=\frac{-3025±\sqrt{9150625-4\left(-32\right)\times 302500}}{2\left(-32\right)}$$

Multiply -4 times -32.

$$x=\frac{-3025±\sqrt{9150625+128\times 302500}}{2\left(-32\right)}$$

Multiply 128 times 302500.

$$x=\frac{-3025±\sqrt{9150625+38720000}}{2\left(-32\right)}$$

Add 9150625 to 38720000.

$$x=\frac{-3025±\sqrt{47870625}}{2\left(-32\right)}$$

Take the square root of 47870625.

$$x=\frac{-3025±275\sqrt{633}}{2\left(-32\right)}$$

Multiply 2 times -32.

$$x=\frac{-3025±275\sqrt{633}}{-64}$$

Now solve the equation x=\frac{-3025±275\sqrt{633}}{-64} when ± is plus. Add -3025 to 275\sqrt{633}.

$$x=\frac{275\sqrt{633}-3025}{-64}$$

Divide -3025+275\sqrt{633} by -64.

$$x=\frac{3025-275\sqrt{633}}{64}$$

Now solve the equation x=\frac{-3025±275\sqrt{633}}{-64} when ± is minus. Subtract 275\sqrt{633} from -3025.

$$x=\frac{-275\sqrt{633}-3025}{-64}$$

Divide -3025-275\sqrt{633} by -64.

$$x=\frac{275\sqrt{633}+3025}{64}$$

The equation is now solved.

$$x=\frac{3025-275\sqrt{633}}{64} x=\frac{275\sqrt{633}+3025}{64}$$

Steps for Completing the Square

Calculate 55 to the power of 2 and get 3025.

$$\frac{-32x^{2}}{3025}+x+200=100$$

Subtract 200 from both sides.

$$\frac{-32x^{2}}{3025}+x=100-200$$

Subtract 200 from 100 to get -100.

$$\frac{-32x^{2}}{3025}+x=-100$$

Multiply both sides of the equation by 3025.

$$-32x^{2}+3025x=-302500$$

Divide both sides by -32.

$$\frac{-32x^{2}+3025x}{-32}=-\frac{302500}{-32}$$

Dividing by -32 undoes the multiplication by -32.

$$x^{2}+\frac{3025}{-32}x=-\frac{302500}{-32}$$

Divide 3025 by -32.

$$x^{2}-\frac{3025}{32}x=-\frac{302500}{-32}$$

Reduce the fraction \frac{-302500}{-32} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{3025}{32}x=\frac{75625}{8}$$

Divide -\frac{3025}{32}, the coefficient of the x term, by 2 to get -\frac{3025}{64}. Then add the square of -\frac{3025}{64} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{3025}{32}x+\left(-\frac{3025}{64}\right)^{2}=\frac{75625}{8}+\left(-\frac{3025}{64}\right)^{2}$$

Square -\frac{3025}{64} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{3025}{32}x+\frac{9150625}{4096}=\frac{75625}{8}+\frac{9150625}{4096}$$

Add \frac{75625}{8} to \frac{9150625}{4096} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{3025}{32}x+\frac{9150625}{4096}=\frac{47870625}{4096}$$

Factor x^{2}-\frac{3025}{32}x+\frac{9150625}{4096}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{3025}{64}\right)^{2}=\frac{47870625}{4096}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{3025}{64}\right)^{2}}=\sqrt{\frac{47870625}{4096}}$$

Simplify.

$$x-\frac{3025}{64}=\frac{275\sqrt{633}}{64} x-\frac{3025}{64}=-\frac{275\sqrt{633}}{64}$$

Add \frac{3025}{64} to both sides of the equation.

$$x=\frac{275\sqrt{633}+3025}{64} x=\frac{3025-275\sqrt{633}}{64}$$