Solve 1=4lambda^2-8(2-1)

$$1 = 4 \lambda ^ { 2 } - 8 ( 2 - 1 )$$

Solve for λ

$$\lambda =\frac{3}{2}=1.5$$

$$\lambda =-\frac{3}{2}=-1.5$$

Steps Using Difference of Squares

Subtract 1 from 2 to get 1.

$$1=4\lambda ^{2}-8\times 1$$

Multiply 8 and 1 to get 8.

$$1=4\lambda ^{2}-8$$

Swap sides so that all variable terms are on the left hand side.

$$4\lambda ^{2}-8=1$$

Subtract 1 from both sides.

$$4\lambda ^{2}-8-1=0$$

Subtract 1 from -8 to get -9.

$$4\lambda ^{2}-9=0$$

Consider 4\lambda ^{2}-9. Rewrite 4\lambda ^{2}-9 as \left(2\lambda \right)^{2}-3^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(2\lambda -3\right)\left(2\lambda +3\right)=0$$

To find equation solutions, solve 2\lambda -3=0 and 2\lambda +3=0.

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

Steps by Finding Square Root

Subtract 1 from 2 to get 1.

$$1=4\lambda ^{2}-8\times 1$$

Multiply 8 and 1 to get 8.

$$1=4\lambda ^{2}-8$$

Swap sides so that all variable terms are on the left hand side.

$$4\lambda ^{2}-8=1$$

Add 8 to both sides.

$$4\lambda ^{2}=1+8$$

Add 1 and 8 to get 9.

$$4\lambda ^{2}=9$$

Divide both sides by 4.

$$\lambda ^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Subtract 1 from 2 to get 1.

$$1=4\lambda ^{2}-8\times 1$$

Multiply 8 and 1 to get 8.

$$1=4\lambda ^{2}-8$$

Swap sides so that all variable terms are on the left hand side.

$$4\lambda ^{2}-8=1$$

Subtract 1 from both sides.

$$4\lambda ^{2}-8-1=0$$

Subtract 1 from -8 to get -9.

$$4\lambda ^{2}-9=0$$

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

$$\lambda =\frac{0±\sqrt{0^{2}-4\times 4\left(-9\right)}}{2\times 4}$$

Square 0.

$$\lambda =\frac{0±\sqrt{-4\times 4\left(-9\right)}}{2\times 4}$$

Multiply -4 times 4.

$$\lambda =\frac{0±\sqrt{-16\left(-9\right)}}{2\times 4}$$

Multiply -16 times -9.

$$\lambda =\frac{0±\sqrt{144}}{2\times 4}$$

Take the square root of 144.

$$\lambda =\frac{0±12}{2\times 4}$$

Multiply 2 times 4.

$$\lambda =\frac{0±12}{8}$$

Now solve the equation \lambda =\frac{0±12}{8} when ± is plus. Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$\lambda =\frac{3}{2}$$

Now solve the equation \lambda =\frac{0±12}{8} when ± is minus. Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

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

The equation is now solved.

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