Solve (x-16)(x-19)=25/64

$$( x - 16 ) ( x - 19 ) = \frac { 25 } { 64 }$$

Solve for x

$$x = \frac{153}{8} = 19\frac{1}{8} = 19.125$$

$$x = \frac{127}{8} = 15\frac{7}{8} = 15.875$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-16 by x-19 and combine like terms.

$$x^{2}-35x+304=\frac{25}{64}$$

Subtract \frac{25}{64} from both sides.

$$x^{2}-35x+304-\frac{25}{64}=0$$

Subtract \frac{25}{64} from 304 to get \frac{19431}{64}.

$$x^{2}-35x+\frac{19431}{64}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -35 for b, and \frac{19431}{64} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times \frac{19431}{64}}}{2}$$

Square -35.

$$x=\frac{-\left(-35\right)±\sqrt{1225-4\times \frac{19431}{64}}}{2}$$

Multiply -4 times \frac{19431}{64}.

$$x=\frac{-\left(-35\right)±\sqrt{1225-\frac{19431}{16}}}{2}$$

Add 1225 to -\frac{19431}{16}.

$$x=\frac{-\left(-35\right)±\sqrt{\frac{169}{16}}}{2}$$

Take the square root of \frac{169}{16}.

$$x=\frac{-\left(-35\right)±\frac{13}{4}}{2}$$

The opposite of -35 is 35.

$$x=\frac{35±\frac{13}{4}}{2}$$

Now solve the equation x=\frac{35±\frac{13}{4}}{2} when ± is plus. Add 35 to \frac{13}{4}.

$$x=\frac{\frac{153}{4}}{2}$$

Divide \frac{153}{4} by 2.

$$x=\frac{153}{8}$$

Now solve the equation x=\frac{35±\frac{13}{4}}{2} when ± is minus. Subtract \frac{13}{4} from 35.

$$x=\frac{\frac{127}{4}}{2}$$

Divide \frac{127}{4} by 2.

$$x=\frac{127}{8}$$

The equation is now solved.

$$x=\frac{153}{8} x=\frac{127}{8}$$

Steps for Completing the Square

Use the distributive property to multiply x-16 by x-19 and combine like terms.

$$x^{2}-35x+304=\frac{25}{64}$$

Subtract 304 from both sides.

$$x^{2}-35x=\frac{25}{64}-304$$

Subtract 304 from \frac{25}{64} to get -\frac{19431}{64}.

$$x^{2}-35x=-\frac{19431}{64}$$

Divide -35, the coefficient of the x term, by 2 to get -\frac{35}{2}. Then add the square of -\frac{35}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-35x+\left(-\frac{35}{2}\right)^{2}=-\frac{19431}{64}+\left(-\frac{35}{2}\right)^{2}$$

Square -\frac{35}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-35x+\frac{1225}{4}=-\frac{19431}{64}+\frac{1225}{4}$$

Add -\frac{19431}{64} to \frac{1225}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-35x+\frac{1225}{4}=\frac{169}{64}$$

Factor x^{2}-35x+\frac{1225}{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}.

$$\left(x-\frac{35}{2}\right)^{2}=\frac{169}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{2}\right)^{2}}=\sqrt{\frac{169}{64}}$$

Simplify.

$$x-\frac{35}{2}=\frac{13}{8} x-\frac{35}{2}=-\frac{13}{8}$$

Add \frac{35}{2} to both sides of the equation.

$$x=\frac{153}{8} x=\frac{127}{8}$$