Solve 11*4(x-1/2)^2=1

$$11 \cdot 4 ( x - \frac { 1 } { 2 } ) ^ { 2 } = 1$$

Solve for x

$$x=\frac{\sqrt{11}}{22}+\frac{1}{2}\approx 0.650755672$$

$$x=-\frac{\sqrt{11}}{22}+\frac{1}{2}\approx 0.349244328$$

Steps Using the Quadratic Formula

Multiply 11 and 4 to get 44.

$$44\left(x-\frac{1}{2}\right)^{2}=1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{2}\right)^{2}.

$$44\left(x^{2}-x+\frac{1}{4}\right)=1$$

Use the distributive property to multiply 44 by x^{2}-x+\frac{1}{4}.

$$44x^{2}-44x+11=1$$

Subtract 1 from both sides.

$$44x^{2}-44x+11-1=0$$

Subtract 1 from 11 to get 10.

$$44x^{2}-44x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 44 for a, -44 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 44\times 10}}{2\times 44}$$

Square -44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-4\times 44\times 10}}{2\times 44}$$

Multiply -4 times 44.

$$x=\frac{-\left(-44\right)±\sqrt{1936-176\times 10}}{2\times 44}$$

Multiply -176 times 10.

$$x=\frac{-\left(-44\right)±\sqrt{1936-1760}}{2\times 44}$$

Add 1936 to -1760.

$$x=\frac{-\left(-44\right)±\sqrt{176}}{2\times 44}$$

Take the square root of 176.

$$x=\frac{-\left(-44\right)±4\sqrt{11}}{2\times 44}$$

The opposite of -44 is 44.

$$x=\frac{44±4\sqrt{11}}{2\times 44}$$

Multiply 2 times 44.

$$x=\frac{44±4\sqrt{11}}{88}$$

Now solve the equation x=\frac{44±4\sqrt{11}}{88} when ± is plus. Add 44 to 4\sqrt{11}.

$$x=\frac{4\sqrt{11}+44}{88}$$

Divide 44+4\sqrt{11} by 88.

$$x=\frac{\sqrt{11}}{22}+\frac{1}{2}$$

Now solve the equation x=\frac{44±4\sqrt{11}}{88} when ± is minus. Subtract 4\sqrt{11} from 44.

$$x=\frac{44-4\sqrt{11}}{88}$$

Divide 44-4\sqrt{11} by 88.

$$x=-\frac{\sqrt{11}}{22}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{11}}{22}+\frac{1}{2} x=-\frac{\sqrt{11}}{22}+\frac{1}{2}$$

Steps for Completing the Square

Multiply 11 and 4 to get 44.

$$44\left(x-\frac{1}{2}\right)^{2}=1$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{2}\right)^{2}.

$$44\left(x^{2}-x+\frac{1}{4}\right)=1$$

Use the distributive property to multiply 44 by x^{2}-x+\frac{1}{4}.

$$44x^{2}-44x+11=1$$

Subtract 11 from both sides.

$$44x^{2}-44x=1-11$$

Subtract 11 from 1 to get -10.

$$44x^{2}-44x=-10$$

Divide both sides by 44.

$$\frac{44x^{2}-44x}{44}=-\frac{10}{44}$$

Dividing by 44 undoes the multiplication by 44.

$$x^{2}+\left(-\frac{44}{44}\right)x=-\frac{10}{44}$$

Divide -44 by 44.

$$x^{2}-x=-\frac{10}{44}$$

Reduce the fraction \frac{-10}{44} to lowest terms by extracting and canceling out 2.

$$x^{2}-x=-\frac{5}{22}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{5}{22}+\left(-\frac{1}{2}\right)^{2}$$

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

$$x^{2}-x+\frac{1}{4}=-\frac{5}{22}+\frac{1}{4}$$

Add -\frac{5}{22} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{1}{44}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{1}{44}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{44}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{11}}{22} x-\frac{1}{2}=-\frac{\sqrt{11}}{22}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{11}}{22}+\frac{1}{2} x=-\frac{\sqrt{11}}{22}+\frac{1}{2}$$