Solve x^2-50x-6000000=0

$${ x }^{ 2 } -50x-6000000=0$$

Solve for x

$$x=25\sqrt{9601}+25\approx 2474.617317052$$

$$x=25-25\sqrt{9601}\approx -2424.617317052$$

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.

$$x^{2}-50x-6000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -50 for b, and -6000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-6000000\right)}}{2}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-6000000\right)}}{2}$$

Multiply -4 times -6000000.

$$x=\frac{-\left(-50\right)±\sqrt{2500+24000000}}{2}$$

Add 2500 to 24000000.

$$x=\frac{-\left(-50\right)±\sqrt{24002500}}{2}$$

Take the square root of 24002500.

$$x=\frac{-\left(-50\right)±50\sqrt{9601}}{2}$$

The opposite of -50 is 50.

$$x=\frac{50±50\sqrt{9601}}{2}$$

Now solve the equation x=\frac{50±50\sqrt{9601}}{2} when ± is plus. Add 50 to 50\sqrt{9601}.

$$x=\frac{50\sqrt{9601}+50}{2}$$

Divide 50+50\sqrt{9601} by 2.

$$x=25\sqrt{9601}+25$$

Now solve the equation x=\frac{50±50\sqrt{9601}}{2} when ± is minus. Subtract 50\sqrt{9601} from 50.

$$x=\frac{50-50\sqrt{9601}}{2}$$

Divide 50-50\sqrt{9601} by 2.

$$x=25-25\sqrt{9601}$$

The equation is now solved.

$$x=25\sqrt{9601}+25 x=25-25\sqrt{9601}$$

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.

$$x^{2}-50x-6000000=0$$

Add 6000000 to both sides of the equation.

$$x^{2}-50x-6000000-\left(-6000000\right)=-\left(-6000000\right)$$

Subtracting -6000000 from itself leaves 0.

$$x^{2}-50x=-\left(-6000000\right)$$

Subtract -6000000 from 0.

$$x^{2}-50x=6000000$$

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.

$$x^{2}-50x+\left(-25\right)^{2}=6000000+\left(-25\right)^{2}$$

Square -25.

$$x^{2}-50x+625=6000000+625$$

Add 6000000 to 625.

$$x^{2}-50x+625=6000625$$

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}.

$$\left(x-25\right)^{2}=6000625$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-25\right)^{2}}=\sqrt{6000625}$$

Simplify.

$$x-25=25\sqrt{9601} x-25=-25\sqrt{9601}$$

Add 25 to both sides of the equation.

$$x=25\sqrt{9601}+25 x=25-25\sqrt{9601}$$