Solve 3x^2+4{left(11/16x-11/16right)}^2-12=0

$$3 { x }^{ 2 } +4 { \left( \frac{ 11 }{ 16 } x- \frac{ 11 }{ 16 } \right) }^{ 2 } -12=0$$

Solve for x

$$x = \frac{24 \sqrt{377} + 121}{313} \approx 1.875385649$$

$$x=\frac{121-24\sqrt{377}}{313}\approx -1.10222271$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{11}{16}x-\frac{11}{16}\right)^{2}.

$$3x^{2}+4\left(\frac{121}{256}x^{2}-\frac{121}{128}x+\frac{121}{256}\right)-12=0$$

Use the distributive property to multiply 4 by \frac{121}{256}x^{2}-\frac{121}{128}x+\frac{121}{256}.

$$3x^{2}+\frac{121}{64}x^{2}-\frac{121}{32}x+\frac{121}{64}-12=0$$

Combine 3x^{2} and \frac{121}{64}x^{2} to get \frac{313}{64}x^{2}.

$$\frac{313}{64}x^{2}-\frac{121}{32}x+\frac{121}{64}-12=0$$

Subtract 12 from \frac{121}{64} to get -\frac{647}{64}.

$$\frac{313}{64}x^{2}-\frac{121}{32}x-\frac{647}{64}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{313}{64} for a, -\frac{121}{32} for b, and -\frac{647}{64} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{121}{32}\right)±\sqrt{\left(-\frac{121}{32}\right)^{2}-4\times \frac{313}{64}\left(-\frac{647}{64}\right)}}{2\times \frac{313}{64}}$$

Square -\frac{121}{32} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{121}{32}\right)±\sqrt{\frac{14641}{1024}-4\times \frac{313}{64}\left(-\frac{647}{64}\right)}}{2\times \frac{313}{64}}$$

Multiply -4 times \frac{313}{64}.

$$x=\frac{-\left(-\frac{121}{32}\right)±\sqrt{\frac{14641}{1024}-\frac{313}{16}\left(-\frac{647}{64}\right)}}{2\times \frac{313}{64}}$$

Multiply -\frac{313}{16} times -\frac{647}{64} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{121}{32}\right)±\sqrt{\frac{14641+202511}{1024}}}{2\times \frac{313}{64}}$$

Add \frac{14641}{1024} to \frac{202511}{1024} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{121}{32}\right)±\sqrt{\frac{3393}{16}}}{2\times \frac{313}{64}}$$

Take the square root of \frac{3393}{16}.

$$x=\frac{-\left(-\frac{121}{32}\right)±\frac{3\sqrt{377}}{4}}{2\times \frac{313}{64}}$$

The opposite of -\frac{121}{32} is \frac{121}{32}.

$$x=\frac{\frac{121}{32}±\frac{3\sqrt{377}}{4}}{2\times \frac{313}{64}}$$

Multiply 2 times \frac{313}{64}.

$$x=\frac{\frac{121}{32}±\frac{3\sqrt{377}}{4}}{\frac{313}{32}}$$

Now solve the equation x=\frac{\frac{121}{32}±\frac{3\sqrt{377}}{4}}{\frac{313}{32}} when ± is plus. Add \frac{121}{32} to \frac{3\sqrt{377}}{4}.

$$x=\frac{\frac{3\sqrt{377}}{4}+\frac{121}{32}}{\frac{313}{32}}$$

Divide \frac{121}{32}+\frac{3\sqrt{377}}{4} by \frac{313}{32} by multiplying \frac{121}{32}+\frac{3\sqrt{377}}{4} by the reciprocal of \frac{313}{32}.

$$x=\frac{24\sqrt{377}+121}{313}$$

Now solve the equation x=\frac{\frac{121}{32}±\frac{3\sqrt{377}}{4}}{\frac{313}{32}} when ± is minus. Subtract \frac{3\sqrt{377}}{4} from \frac{121}{32}.

$$x=\frac{-\frac{3\sqrt{377}}{4}+\frac{121}{32}}{\frac{313}{32}}$$

Divide \frac{121}{32}-\frac{3\sqrt{377}}{4} by \frac{313}{32} by multiplying \frac{121}{32}-\frac{3\sqrt{377}}{4} by the reciprocal of \frac{313}{32}.

$$x=\frac{121-24\sqrt{377}}{313}$$

The equation is now solved.

$$x=\frac{24\sqrt{377}+121}{313} x=\frac{121-24\sqrt{377}}{313}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{11}{16}x-\frac{11}{16}\right)^{2}.

$$3x^{2}+4\left(\frac{121}{256}x^{2}-\frac{121}{128}x+\frac{121}{256}\right)-12=0$$

Use the distributive property to multiply 4 by \frac{121}{256}x^{2}-\frac{121}{128}x+\frac{121}{256}.

$$3x^{2}+\frac{121}{64}x^{2}-\frac{121}{32}x+\frac{121}{64}-12=0$$

Combine 3x^{2} and \frac{121}{64}x^{2} to get \frac{313}{64}x^{2}.

$$\frac{313}{64}x^{2}-\frac{121}{32}x+\frac{121}{64}-12=0$$

Subtract 12 from \frac{121}{64} to get -\frac{647}{64}.

$$\frac{313}{64}x^{2}-\frac{121}{32}x-\frac{647}{64}=0$$

Add \frac{647}{64} to both sides. Anything plus zero gives itself.

$$\frac{313}{64}x^{2}-\frac{121}{32}x=\frac{647}{64}$$

Divide both sides of the equation by \frac{313}{64}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{313}{64}x^{2}-\frac{121}{32}x}{\frac{313}{64}}=\frac{\frac{647}{64}}{\frac{313}{64}}$$

Dividing by \frac{313}{64} undoes the multiplication by \frac{313}{64}.

$$x^{2}+\left(-\frac{\frac{121}{32}}{\frac{313}{64}}\right)x=\frac{\frac{647}{64}}{\frac{313}{64}}$$

Divide -\frac{121}{32} by \frac{313}{64} by multiplying -\frac{121}{32} by the reciprocal of \frac{313}{64}.

$$x^{2}-\frac{242}{313}x=\frac{\frac{647}{64}}{\frac{313}{64}}$$

Divide \frac{647}{64} by \frac{313}{64} by multiplying \frac{647}{64} by the reciprocal of \frac{313}{64}.

$$x^{2}-\frac{242}{313}x=\frac{647}{313}$$

Divide -\frac{242}{313}, the coefficient of the x term, by 2 to get -\frac{121}{313}. Then add the square of -\frac{121}{313} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{242}{313}x+\left(-\frac{121}{313}\right)^{2}=\frac{647}{313}+\left(-\frac{121}{313}\right)^{2}$$

Square -\frac{121}{313} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{242}{313}x+\frac{14641}{97969}=\frac{647}{313}+\frac{14641}{97969}$$

Add \frac{647}{313} to \frac{14641}{97969} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{242}{313}x+\frac{14641}{97969}=\frac{217152}{97969}$$

Factor x^{2}-\frac{242}{313}x+\frac{14641}{97969}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{121}{313}\right)^{2}=\frac{217152}{97969}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{121}{313}\right)^{2}}=\sqrt{\frac{217152}{97969}}$$

Simplify.

$$x-\frac{121}{313}=\frac{24\sqrt{377}}{313} x-\frac{121}{313}=-\frac{24\sqrt{377}}{313}$$

Add \frac{121}{313} to both sides of the equation.

$$x=\frac{24\sqrt{377}+121}{313} x=\frac{121-24\sqrt{377}}{313}$$