Solve (x-1/2)(x+1/2)=1997

$$( \frac{ x-1 }{ 2 } )( \frac{ x+1 }{ 2 } )=1997$$

Solve for x

$$x=\sqrt{7989}\approx 89.381206078$$

$$x=-\sqrt{7989}\approx -89.381206078$$

Steps by Finding Square Root

Multiply both sides of the equation by 2.

$$\left(x-1\right)\times \frac{x+1}{2}=3994$$

Express \left(x-1\right)\times \frac{x+1}{2} as a single fraction.

$$\frac{\left(x-1\right)\left(x+1\right)}{2}=3994$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}-1^{2}}{2}=3994$$

Calculate 1 to the power of 2 and get 1.

$$\frac{x^{2}-1}{2}=3994$$

Divide each term of x^{2}-1 by 2 to get \frac{1}{2}x^{2}-\frac{1}{2}.

$$\frac{1}{2}x^{2}-\frac{1}{2}=3994$$

Add \frac{1}{2} to both sides.

$$\frac{1}{2}x^{2}=3994+\frac{1}{2}$$

Convert 3994 to fraction \frac{7988}{2}.

$$\frac{1}{2}x^{2}=\frac{7988}{2}+\frac{1}{2}$$

Since \frac{7988}{2} and \frac{1}{2} have the same denominator, add them by adding their numerators.

$$\frac{1}{2}x^{2}=\frac{7988+1}{2}$$

Add 7988 and 1 to get 7989.

$$\frac{1}{2}x^{2}=\frac{7989}{2}$$

Multiply both sides by 2, the reciprocal of \frac{1}{2}.

$$x^{2}=\frac{7989}{2}\times 2$$

Cancel out 2 and 2.

$$x^{2}=7989$$

Take the square root of both sides of the equation.

$$x=\sqrt{7989} x=-\sqrt{7989}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$\left(x-1\right)\times \frac{x+1}{2}=3994$$

Express \left(x-1\right)\times \frac{x+1}{2} as a single fraction.

$$\frac{\left(x-1\right)\left(x+1\right)}{2}=3994$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{x^{2}-1^{2}}{2}=3994$$

Calculate 1 to the power of 2 and get 1.

$$\frac{x^{2}-1}{2}=3994$$

Divide each term of x^{2}-1 by 2 to get \frac{1}{2}x^{2}-\frac{1}{2}.

$$\frac{1}{2}x^{2}-\frac{1}{2}=3994$$

Subtract 3994 from both sides.

$$\frac{1}{2}x^{2}-\frac{1}{2}-3994=0$$

Convert 3994 to fraction \frac{7988}{2}.

$$\frac{1}{2}x^{2}-\frac{1}{2}-\frac{7988}{2}=0$$

Since -\frac{1}{2} and \frac{7988}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{2}x^{2}+\frac{-1-7988}{2}=0$$

Subtract 7988 from -1 to get -7989.

$$\frac{1}{2}x^{2}-\frac{7989}{2}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 0 for b, and -\frac{7989}{2} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times \frac{1}{2}\left(-\frac{7989}{2}\right)}}{2\times \frac{1}{2}}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times \frac{1}{2}\left(-\frac{7989}{2}\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$x=\frac{0±\sqrt{-2\left(-\frac{7989}{2}\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -\frac{7989}{2}.

$$x=\frac{0±\sqrt{7989}}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{0±\sqrt{7989}}{1}$$

Now solve the equation x=\frac{0±\sqrt{7989}}{1} when ± is plus.

$$x=\sqrt{7989}$$

Now solve the equation x=\frac{0±\sqrt{7989}}{1} when ± is minus.

$$x=-\sqrt{7989}$$

The equation is now solved.

$$x=\sqrt{7989} x=-\sqrt{7989}$$