Solve 200=25.98x+4.9x^2

$$200 = 25.98x+4.9 { x }^{ 2 }$$

Solve for x

$$x = \frac{\sqrt{11487401} - 1299}{490} \approx 4.26593237$$

$$x=\frac{-\sqrt{11487401}-1299}{490}\approx -9.567973186$$

Steps Using the Quadratic Formula

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

$$25.98x+4.9x^{2}=200$$

Subtract 200 from both sides.

$$25.98x+4.9x^{2}-200=0$$

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.

$$4.9x^{2}+25.98x-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4.9 for a, 25.98 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-25.98±\sqrt{25.98^{2}-4\times 4.9\left(-200\right)}}{2\times 4.9}$$

Square 25.98 by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-25.98±\sqrt{674.9604-4\times 4.9\left(-200\right)}}{2\times 4.9}$$

Multiply -4 times 4.9.

$$x=\frac{-25.98±\sqrt{674.9604-19.6\left(-200\right)}}{2\times 4.9}$$

Multiply -19.6 times -200.

$$x=\frac{-25.98±\sqrt{674.9604+3920}}{2\times 4.9}$$

Add 674.9604 to 3920.

$$x=\frac{-25.98±\sqrt{4594.9604}}{2\times 4.9}$$

Take the square root of 4594.9604.

$$x=\frac{-25.98±\frac{\sqrt{11487401}}{50}}{2\times 4.9}$$

Multiply 2 times 4.9.

$$x=\frac{-25.98±\frac{\sqrt{11487401}}{50}}{9.8}$$

Now solve the equation x=\frac{-25.98±\frac{\sqrt{11487401}}{50}}{9.8} when ± is plus. Add -25.98 to \frac{\sqrt{11487401}}{50}.

$$x=\frac{\sqrt{11487401}-1299}{9.8\times 50}$$

Divide \frac{-1299+\sqrt{11487401}}{50} by 9.8 by multiplying \frac{-1299+\sqrt{11487401}}{50} by the reciprocal of 9.8.

$$x=\frac{\sqrt{11487401}-1299}{490}$$

Now solve the equation x=\frac{-25.98±\frac{\sqrt{11487401}}{50}}{9.8} when ± is minus. Subtract \frac{\sqrt{11487401}}{50} from -25.98.

$$x=\frac{-\sqrt{11487401}-1299}{9.8\times 50}$$

Divide \frac{-1299-\sqrt{11487401}}{50} by 9.8 by multiplying \frac{-1299-\sqrt{11487401}}{50} by the reciprocal of 9.8.

$$x=\frac{-\sqrt{11487401}-1299}{490}$$

The equation is now solved.

$$x=\frac{\sqrt{11487401}-1299}{490} x=\frac{-\sqrt{11487401}-1299}{490}$$

Steps for Completing the Square

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

$$25.98x+4.9x^{2}=200$$

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.

$$4.9x^{2}+25.98x=200$$

Divide both sides of the equation by 4.9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4.9x^{2}+25.98x}{4.9}=\frac{200}{4.9}$$

Dividing by 4.9 undoes the multiplication by 4.9.

$$x^{2}+\frac{25.98}{4.9}x=\frac{200}{4.9}$$

Divide 25.98 by 4.9 by multiplying 25.98 by the reciprocal of 4.9.

$$x^{2}+\frac{1299}{245}x=\frac{200}{4.9}$$

Divide 200 by 4.9 by multiplying 200 by the reciprocal of 4.9.

$$x^{2}+\frac{1299}{245}x=\frac{2000}{49}$$

Divide \frac{1299}{245}, the coefficient of the x term, by 2 to get \frac{1299}{490}. Then add the square of \frac{1299}{490} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1299}{245}x+\frac{1299}{490}^{2}=\frac{2000}{49}+\frac{1299}{490}^{2}$$

Square \frac{1299}{490} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1299}{245}x+\frac{1687401}{240100}=\frac{2000}{49}+\frac{1687401}{240100}$$

Add \frac{2000}{49} to \frac{1687401}{240100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1299}{245}x+\frac{1687401}{240100}=\frac{11487401}{240100}$$

Factor x^{2}+\frac{1299}{245}x+\frac{1687401}{240100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1299}{490}\right)^{2}=\frac{11487401}{240100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1299}{490}\right)^{2}}=\sqrt{\frac{11487401}{240100}}$$

Simplify.

$$x+\frac{1299}{490}=\frac{\sqrt{11487401}}{490} x+\frac{1299}{490}=-\frac{\sqrt{11487401}}{490}$$

Subtract \frac{1299}{490} from both sides of the equation.

$$x=\frac{\sqrt{11487401}-1299}{490} x=\frac{-\sqrt{11487401}-1299}{490}$$