Solve 4x^2-200x+2.5=0

$$4 { x }^{ 2 } -200x+2.5=0$$

Solve for x

$$x=\frac{3\sqrt{1110}}{4}+25\approx 49.987496873$$

$$x=-\frac{3\sqrt{1110}}{4}+25\approx 0.012503127$$

Steps Using the Quadratic Formula

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.

$$4x^{2}-200x+2.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -200 for b, and 2.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\times 4\times 2.5}}{2\times 4}$$

Square -200.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\times 4\times 2.5}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-200\right)±\sqrt{40000-16\times 2.5}}{2\times 4}$$

Multiply -16 times 2.5.

$$x=\frac{-\left(-200\right)±\sqrt{40000-40}}{2\times 4}$$

Add 40000 to -40.

$$x=\frac{-\left(-200\right)±\sqrt{39960}}{2\times 4}$$

Take the square root of 39960.

$$x=\frac{-\left(-200\right)±6\sqrt{1110}}{2\times 4}$$

The opposite of -200 is 200.

$$x=\frac{200±6\sqrt{1110}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{200±6\sqrt{1110}}{8}$$

Now solve the equation x=\frac{200±6\sqrt{1110}}{8} when ± is plus. Add 200 to 6\sqrt{1110}.

$$x=\frac{6\sqrt{1110}+200}{8}$$

Divide 200+6\sqrt{1110} by 8.

$$x=\frac{3\sqrt{1110}}{4}+25$$

Now solve the equation x=\frac{200±6\sqrt{1110}}{8} when ± is minus. Subtract 6\sqrt{1110} from 200.

$$x=\frac{200-6\sqrt{1110}}{8}$$

Divide 200-6\sqrt{1110} by 8.

$$x=-\frac{3\sqrt{1110}}{4}+25$$

The equation is now solved.

$$x=\frac{3\sqrt{1110}}{4}+25 x=-\frac{3\sqrt{1110}}{4}+25$$

Steps for Completing the Square

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.

$$4x^{2}-200x+2.5=0$$

Subtract 2.5 from both sides of the equation.

$$4x^{2}-200x+2.5-2.5=-2.5$$

Subtracting 2.5 from itself leaves 0.

$$4x^{2}-200x=-2.5$$

Divide both sides by 4.

$$\frac{4x^{2}-200x}{4}=-\frac{2.5}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{200}{4}\right)x=-\frac{2.5}{4}$$

Divide -200 by 4.

$$x^{2}-50x=-\frac{2.5}{4}$$

Divide -2.5 by 4.

$$x^{2}-50x=-0.625$$

Divide -50, the coefficient of the x term, by 2 to get -25. Then add the square of -25 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-50x+\left(-25\right)^{2}=-0.625+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=-0.625+625$$

Add -0.625 to 625.

$$x^{2}-50x+625=624.375$$

Factor x^{2}-50x+625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-25\right)^{2}=624.375$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{624.375}$$

Simplify.

$$x-25=\frac{3\sqrt{1110}}{4} x-25=-\frac{3\sqrt{1110}}{4}$$

Add 25 to both sides of the equation.

$$x=\frac{3\sqrt{1110}}{4}+25 x=-\frac{3\sqrt{1110}}{4}+25$$