Solve (49-x)*(x+1+49)=(49div2)(1+50)

$$(49-x) \times (x+1+49)=(49 \div 2)(1+50)$$

Solve for x

$$x = \frac{\sqrt{4803} - 1}{2} \approx 34.151839778$$

$$x=\frac{-\sqrt{4803}-1}{2}\approx -35.151839778$$

Steps Using the Quadratic Formula

Add 1 and 49 to get 50.

$$\left(49-x\right)\left(x+50\right)=\frac{49}{2}\left(1+50\right)$$

Use the distributive property to multiply 49-x by x+50 and combine like terms.

$$-x+2450-x^{2}=\frac{49}{2}\left(1+50\right)$$

Add 1 and 50 to get 51.

$$-x+2450-x^{2}=\frac{49}{2}\times 51$$

Multiply \frac{49}{2} and 51 to get \frac{2499}{2}.

$$-x+2450-x^{2}=\frac{2499}{2}$$

Subtract \frac{2499}{2} from both sides.

$$-x+2450-x^{2}-\frac{2499}{2}=0$$

Subtract \frac{2499}{2} from 2450 to get \frac{2401}{2}.

$$-x+\frac{2401}{2}-x^{2}=0$$

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.

$$-x^{2}-x+\frac{2401}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and \frac{2401}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times \frac{2401}{2}}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times \frac{2401}{2}}}{2\left(-1\right)}$$

Multiply 4 times \frac{2401}{2}.

$$x=\frac{-\left(-1\right)±\sqrt{1+4802}}{2\left(-1\right)}$$

Add 1 to 4802.

$$x=\frac{-\left(-1\right)±\sqrt{4803}}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±\sqrt{4803}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±\sqrt{4803}}{-2}$$

Now solve the equation x=\frac{1±\sqrt{4803}}{-2} when ± is plus. Add 1 to \sqrt{4803}.

$$x=\frac{\sqrt{4803}+1}{-2}$$

Divide 1+\sqrt{4803} by -2.

$$x=\frac{-\sqrt{4803}-1}{2}$$

Now solve the equation x=\frac{1±\sqrt{4803}}{-2} when ± is minus. Subtract \sqrt{4803} from 1.

$$x=\frac{1-\sqrt{4803}}{-2}$$

Divide 1-\sqrt{4803} by -2.

$$x=\frac{\sqrt{4803}-1}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{4803}-1}{2} x=\frac{\sqrt{4803}-1}{2}$$

Steps for Completing the Square

Add 1 and 49 to get 50.

$$\left(49-x\right)\left(x+50\right)=\frac{49}{2}\left(1+50\right)$$

Use the distributive property to multiply 49-x by x+50 and combine like terms.

$$-x+2450-x^{2}=\frac{49}{2}\left(1+50\right)$$

Add 1 and 50 to get 51.

$$-x+2450-x^{2}=\frac{49}{2}\times 51$$

Multiply \frac{49}{2} and 51 to get \frac{2499}{2}.

$$-x+2450-x^{2}=\frac{2499}{2}$$

Subtract 2450 from both sides.

$$-x-x^{2}=\frac{2499}{2}-2450$$

Subtract 2450 from \frac{2499}{2} to get -\frac{2401}{2}.

$$-x-x^{2}=-\frac{2401}{2}$$

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.

$$-x^{2}-x=-\frac{2401}{2}$$

Divide both sides by -1.

$$\frac{-x^{2}-x}{-1}=-\frac{\frac{2401}{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{1}{-1}\right)x=-\frac{\frac{2401}{2}}{-1}$$

Divide -1 by -1.

$$x^{2}+x=-\frac{\frac{2401}{2}}{-1}$$

Divide -\frac{2401}{2} by -1.

$$x^{2}+x=\frac{2401}{2}$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{2401}{2}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=\frac{2401}{2}+\frac{1}{4}$$

Add \frac{2401}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{4803}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{4803}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{4803}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{4803}}{2} x+\frac{1}{2}=-\frac{\sqrt{4803}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{4803}-1}{2} x=\frac{-\sqrt{4803}-1}{2}$$