Solve -x^2+79-50x=0

$$- x ^ { 2 } + 79 - 50 x = 0$$

Solve for x

$$x=8\sqrt{11}-25\approx 1.532998323$$

$$x=-8\sqrt{11}-25\approx -51.532998323$$

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.

$$-x^{2}-50x+79=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -50 for b, and 79 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-50\right)±\sqrt{\left(-50\right)^{2}-4\left(-1\right)\times 79}}{2\left(-1\right)}$$

Square -50.

$$x=\frac{-\left(-50\right)±\sqrt{2500-4\left(-1\right)\times 79}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-50\right)±\sqrt{2500+4\times 79}}{2\left(-1\right)}$$

Multiply 4 times 79.

$$x=\frac{-\left(-50\right)±\sqrt{2500+316}}{2\left(-1\right)}$$

Add 2500 to 316.

$$x=\frac{-\left(-50\right)±\sqrt{2816}}{2\left(-1\right)}$$

Take the square root of 2816.

$$x=\frac{-\left(-50\right)±16\sqrt{11}}{2\left(-1\right)}$$

The opposite of -50 is 50.

$$x=\frac{50±16\sqrt{11}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{50±16\sqrt{11}}{-2}$$

Now solve the equation x=\frac{50±16\sqrt{11}}{-2} when ± is plus. Add 50 to 16\sqrt{11}.

$$x=\frac{16\sqrt{11}+50}{-2}$$

Divide 50+16\sqrt{11} by -2.

$$x=-8\sqrt{11}-25$$

Now solve the equation x=\frac{50±16\sqrt{11}}{-2} when ± is minus. Subtract 16\sqrt{11} from 50.

$$x=\frac{50-16\sqrt{11}}{-2}$$

Divide 50-16\sqrt{11} by -2.

$$x=8\sqrt{11}-25$$

The equation is now solved.

$$x=-8\sqrt{11}-25 x=8\sqrt{11}-25$$

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.

$$-x^{2}-50x+79=0$$

Subtract 79 from both sides of the equation.

$$-x^{2}-50x+79-79=-79$$

Subtracting 79 from itself leaves 0.

$$-x^{2}-50x=-79$$

Divide both sides by -1.

$$\frac{-x^{2}-50x}{-1}=-\frac{79}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{50}{-1}\right)x=-\frac{79}{-1}$$

Divide -50 by -1.

$$x^{2}+50x=-\frac{79}{-1}$$

Divide -79 by -1.

$$x^{2}+50x=79$$

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.

$$x^{2}+50x+25^{2}=79+25^{2}$$

Square 25.

$$x^{2}+50x+625=79+625$$

Add 79 to 625.

$$x^{2}+50x+625=704$$

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}.

$$\left(x+25\right)^{2}=704$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+25\right)^{2}}=\sqrt{704}$$

Simplify.

$$x+25=8\sqrt{11} x+25=-8\sqrt{11}$$

Subtract 25 from both sides of the equation.

$$x=8\sqrt{11}-25 x=-8\sqrt{11}-25$$