Solve 5/4x^2-1/2x+0.25-6.5^2=0

$$\frac{ 5 }{ 4 } { x }^{ 2 } - \frac{ 1 }{ 2 } x+0.25- { 6.5 }^{ 2 } =0$$

Solve for x

$$x=-5.6$$

$$x=6$$

Steps Using the Quadratic Formula

Calculate 6.5 to the power of 2 and get 42.25.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0.25-42.25=0$$

Subtract 42.25 from 0.25 to get -42.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-42=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{5}{4} for a, -\frac{1}{2} for b, and -42 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\left(-\frac{1}{2}\right)^{2}-4\times \frac{5}{4}\left(-42\right)}}{2\times \frac{5}{4}}$$

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

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-4\times \frac{5}{4}\left(-42\right)}}{2\times \frac{5}{4}}$$

Multiply -4 times \frac{5}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}-5\left(-42\right)}}{2\times \frac{5}{4}}$$

Multiply -5 times -42.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{1}{4}+210}}{2\times \frac{5}{4}}$$

Add \frac{1}{4} to 210.

$$x=\frac{-\left(-\frac{1}{2}\right)±\sqrt{\frac{841}{4}}}{2\times \frac{5}{4}}$$

Take the square root of \frac{841}{4}.

$$x=\frac{-\left(-\frac{1}{2}\right)±\frac{29}{2}}{2\times \frac{5}{4}}$$

The opposite of -\frac{1}{2} is \frac{1}{2}.

$$x=\frac{\frac{1}{2}±\frac{29}{2}}{2\times \frac{5}{4}}$$

Multiply 2 times \frac{5}{4}.

$$x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}}$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}} when ± is plus. Add \frac{1}{2} to \frac{29}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{15}{\frac{5}{2}}$$

Divide 15 by \frac{5}{2} by multiplying 15 by the reciprocal of \frac{5}{2}.

$$x=6$$

Now solve the equation x=\frac{\frac{1}{2}±\frac{29}{2}}{\frac{5}{2}} when ± is minus. Subtract \frac{29}{2} from \frac{1}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$x=-\frac{14}{\frac{5}{2}}$$

Divide -14 by \frac{5}{2} by multiplying -14 by the reciprocal of \frac{5}{2}.

$$x=-\frac{28}{5}$$

The equation is now solved.

$$x=6 x=-\frac{28}{5}$$

Steps for Completing the Square

Calculate 6.5 to the power of 2 and get 42.25.

$$\frac{5}{4}x^{2}-\frac{1}{2}x+0.25-42.25=0$$

Subtract 42.25 from 0.25 to get -42.

$$\frac{5}{4}x^{2}-\frac{1}{2}x-42=0$$

Add 42 to both sides. Anything plus zero gives itself.

$$\frac{5}{4}x^{2}-\frac{1}{2}x=42$$

Divide both sides of the equation by \frac{5}{4}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{5}{4}x^{2}-\frac{1}{2}x}{\frac{5}{4}}=\frac{42}{\frac{5}{4}}$$

Dividing by \frac{5}{4} undoes the multiplication by \frac{5}{4}.

$$x^{2}+\left(-\frac{\frac{1}{2}}{\frac{5}{4}}\right)x=\frac{42}{\frac{5}{4}}$$

Divide -\frac{1}{2} by \frac{5}{4} by multiplying -\frac{1}{2} by the reciprocal of \frac{5}{4}.

$$x^{2}-\frac{2}{5}x=\frac{42}{\frac{5}{4}}$$

Divide 42 by \frac{5}{4} by multiplying 42 by the reciprocal of \frac{5}{4}.

$$x^{2}-\frac{2}{5}x=\frac{168}{5}$$

Divide -\frac{2}{5}, the coefficient of the x term, by 2 to get -\frac{1}{5}. Then add the square of -\frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{5}x+\left(-\frac{1}{5}\right)^{2}=\frac{168}{5}+\left(-\frac{1}{5}\right)^{2}$$

Square -\frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{168}{5}+\frac{1}{25}$$

Add \frac{168}{5} to \frac{1}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{5}x+\frac{1}{25}=\frac{841}{25}$$

Factor x^{2}-\frac{2}{5}x+\frac{1}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{5}\right)^{2}=\frac{841}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{5}\right)^{2}}=\sqrt{\frac{841}{25}}$$

Simplify.

$$x-\frac{1}{5}=\frac{29}{5} x-\frac{1}{5}=-\frac{29}{5}$$

Add \frac{1}{5} to both sides of the equation.

$$x=6 x=-\frac{28}{5}$$