Solve -16K^2+164K^2-36K-247/112=0

$$- 16 K ^ { 2 } + 164 K ^ { 2 } - 36 K - \frac { 247 } { 112 } = 0$$

Solve for K

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74}\approx 0.293937844$$

$$K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}\approx -0.050694601$$

Steps Using the Quadratic Formula

Combine -16K^{2} and 164K^{2} to get 148K^{2}.

$$148K^{2}-36K-\frac{247}{112}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 148 for a, -36 for b, and -\frac{247}{112} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$K=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 148\left(-\frac{247}{112}\right)}}{2\times 148}$$

Square -36.

$$K=\frac{-\left(-36\right)±\sqrt{1296-4\times 148\left(-\frac{247}{112}\right)}}{2\times 148}$$

Multiply -4 times 148.

$$K=\frac{-\left(-36\right)±\sqrt{1296-592\left(-\frac{247}{112}\right)}}{2\times 148}$$

Multiply -592 times -\frac{247}{112}.

$$K=\frac{-\left(-36\right)±\sqrt{1296+\frac{9139}{7}}}{2\times 148}$$

Add 1296 to \frac{9139}{7}.

$$K=\frac{-\left(-36\right)±\sqrt{\frac{18211}{7}}}{2\times 148}$$

Take the square root of \frac{18211}{7}.

$$K=\frac{-\left(-36\right)±\frac{\sqrt{127477}}{7}}{2\times 148}$$

The opposite of -36 is 36.

$$K=\frac{36±\frac{\sqrt{127477}}{7}}{2\times 148}$$

Multiply 2 times 148.

$$K=\frac{36±\frac{\sqrt{127477}}{7}}{296}$$

Now solve the equation K=\frac{36±\frac{\sqrt{127477}}{7}}{296} when ± is plus. Add 36 to \frac{\sqrt{127477}}{7}.

$$K=\frac{\frac{\sqrt{127477}}{7}+36}{296}$$

Divide 36+\frac{\sqrt{127477}}{7} by 296.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

Now solve the equation K=\frac{36±\frac{\sqrt{127477}}{7}}{296} when ± is minus. Subtract \frac{\sqrt{127477}}{7} from 36.

$$K=\frac{-\frac{\sqrt{127477}}{7}+36}{296}$$

Divide 36-\frac{\sqrt{127477}}{7} by 296.

$$K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

The equation is now solved.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74} K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$

Steps for Completing the Square

Combine -16K^{2} and 164K^{2} to get 148K^{2}.

$$148K^{2}-36K-\frac{247}{112}=0$$

Add \frac{247}{112} to both sides. Anything plus zero gives itself.

$$148K^{2}-36K=\frac{247}{112}$$

Divide both sides by 148.

$$\frac{148K^{2}-36K}{148}=\frac{\frac{247}{112}}{148}$$

Dividing by 148 undoes the multiplication by 148.

$$K^{2}+\left(-\frac{36}{148}\right)K=\frac{\frac{247}{112}}{148}$$

Reduce the fraction \frac{-36}{148} to lowest terms by extracting and canceling out 4.

$$K^{2}-\frac{9}{37}K=\frac{\frac{247}{112}}{148}$$

Divide \frac{247}{112} by 148.

$$K^{2}-\frac{9}{37}K=\frac{247}{16576}$$

Divide -\frac{9}{37}, the coefficient of the x term, by 2 to get -\frac{9}{74}. Then add the square of -\frac{9}{74} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$K^{2}-\frac{9}{37}K+\left(-\frac{9}{74}\right)^{2}=\frac{247}{16576}+\left(-\frac{9}{74}\right)^{2}$$

Square -\frac{9}{74} by squaring both the numerator and the denominator of the fraction.

$$K^{2}-\frac{9}{37}K+\frac{81}{5476}=\frac{247}{16576}+\frac{81}{5476}$$

Add \frac{247}{16576} to \frac{81}{5476} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$K^{2}-\frac{9}{37}K+\frac{81}{5476}=\frac{18211}{613312}$$

Factor K^{2}-\frac{9}{37}K+\frac{81}{5476}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(K-\frac{9}{74}\right)^{2}=\frac{18211}{613312}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(K-\frac{9}{74}\right)^{2}}=\sqrt{\frac{18211}{613312}}$$

Simplify.

$$K-\frac{9}{74}=\frac{\sqrt{127477}}{2072} K-\frac{9}{74}=-\frac{\sqrt{127477}}{2072}$$

Add \frac{9}{74} to both sides of the equation.

$$K=\frac{\sqrt{127477}}{2072}+\frac{9}{74} K=-\frac{\sqrt{127477}}{2072}+\frac{9}{74}$$