Solve (3lambda+1)^2=1^2+1

$$( 3 \lambda + 1 ) ^ { 2 } = 1 ^ { 2 } + 1$$

Solve for λ

$$\lambda =\frac{\sqrt{2}-1}{3}\approx 0.138071187$$

$$\lambda =\frac{-\sqrt{2}-1}{3}\approx -0.804737854$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\lambda +1\right)^{2}.

$$9\lambda ^{2}+6\lambda +1=1^{2}+1$$

Calculate 1 to the power of 2 and get 1.

$$9\lambda ^{2}+6\lambda +1=1+1$$

Add 1 and 1 to get 2.

$$9\lambda ^{2}+6\lambda +1=2$$

Subtract 2 from both sides.

$$9\lambda ^{2}+6\lambda +1-2=0$$

Subtract 2 from 1 to get -1.

$$9\lambda ^{2}+6\lambda -1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 6 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$\lambda =\frac{-6±\sqrt{6^{2}-4\times 9\left(-1\right)}}{2\times 9}$$

Square 6.

$$\lambda =\frac{-6±\sqrt{36-4\times 9\left(-1\right)}}{2\times 9}$$

Multiply -4 times 9.

$$\lambda =\frac{-6±\sqrt{36-36\left(-1\right)}}{2\times 9}$$

Multiply -36 times -1.

$$\lambda =\frac{-6±\sqrt{36+36}}{2\times 9}$$

Add 36 to 36.

$$\lambda =\frac{-6±\sqrt{72}}{2\times 9}$$

Take the square root of 72.

$$\lambda =\frac{-6±6\sqrt{2}}{2\times 9}$$

Multiply 2 times 9.

$$\lambda =\frac{-6±6\sqrt{2}}{18}$$

Now solve the equation \lambda =\frac{-6±6\sqrt{2}}{18} when ± is plus. Add -6 to 6\sqrt{2}.

$$\lambda =\frac{6\sqrt{2}-6}{18}$$

Divide -6+6\sqrt{2} by 18.

$$\lambda =\frac{\sqrt{2}-1}{3}$$

Now solve the equation \lambda =\frac{-6±6\sqrt{2}}{18} when ± is minus. Subtract 6\sqrt{2} from -6.

$$\lambda =\frac{-6\sqrt{2}-6}{18}$$

Divide -6-6\sqrt{2} by 18.

$$\lambda =\frac{-\sqrt{2}-1}{3}$$

The equation is now solved.

$$\lambda =\frac{\sqrt{2}-1}{3} \lambda =\frac{-\sqrt{2}-1}{3}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3\lambda +1\right)^{2}.

$$9\lambda ^{2}+6\lambda +1=1^{2}+1$$

Calculate 1 to the power of 2 and get 1.

$$9\lambda ^{2}+6\lambda +1=1+1$$

Add 1 and 1 to get 2.

$$9\lambda ^{2}+6\lambda +1=2$$

Subtract 1 from both sides.

$$9\lambda ^{2}+6\lambda =2-1$$

Subtract 1 from 2 to get 1.

$$9\lambda ^{2}+6\lambda =1$$

Divide both sides by 9.

$$\frac{9\lambda ^{2}+6\lambda }{9}=\frac{1}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$\lambda ^{2}+\frac{6}{9}\lambda =\frac{1}{9}$$

Reduce the fraction \frac{6}{9} to lowest terms by extracting and canceling out 3.

$$\lambda ^{2}+\frac{2}{3}\lambda =\frac{1}{9}$$

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$\lambda ^{2}+\frac{2}{3}\lambda +\left(\frac{1}{3}\right)^{2}=\frac{1}{9}+\left(\frac{1}{3}\right)^{2}$$

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$\lambda ^{2}+\frac{2}{3}\lambda +\frac{1}{9}=\frac{1+1}{9}$$

Add \frac{1}{9} to \frac{1}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$\lambda ^{2}+\frac{2}{3}\lambda +\frac{1}{9}=\frac{2}{9}$$

Factor \lambda ^{2}+\frac{2}{3}\lambda +\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(\lambda +\frac{1}{3}\right)^{2}=\frac{2}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(\lambda +\frac{1}{3}\right)^{2}}=\sqrt{\frac{2}{9}}$$

Simplify.

$$\lambda +\frac{1}{3}=\frac{\sqrt{2}}{3} \lambda +\frac{1}{3}=-\frac{\sqrt{2}}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$\lambda =\frac{\sqrt{2}-1}{3} \lambda =\frac{-\sqrt{2}-1}{3}$$