Solve 2lambda^2-2lambda-19leq0

$$2 \lambda ^ { 2 } - 2 \lambda - 19 \leq 0$$

Solve for λ

$$\lambda \in \begin{bmatrix}\frac{1-\sqrt{39}}{2},\frac{\sqrt{39}+1}{2}\end{bmatrix}$$

Steps Using the Quadratic Formula

To solve the inequality, factor the left hand side. Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$2\lambda ^{2}-2\lambda -19=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Substitute 2 for a, -2 for b, and -19 for c in the quadratic formula.

$$\lambda =\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-19\right)}}{2\times 2}$$

Do the calculations.

$$\lambda =\frac{2±2\sqrt{39}}{4}$$

Solve the equation \lambda =\frac{2±2\sqrt{39}}{4} when ± is plus and when ± is minus.

$$\lambda =\frac{\sqrt{39}+1}{2} \lambda =\frac{1-\sqrt{39}}{2}$$

Rewrite the inequality by using the obtained solutions.

$$2\left(\lambda -\frac{\sqrt{39}+1}{2}\right)\left(\lambda -\frac{1-\sqrt{39}}{2}\right)\leq 0$$

For the product to be ≤0, one of the values \lambda -\frac{\sqrt{39}+1}{2} and \lambda -\frac{1-\sqrt{39}}{2} has to be ≥0 and the other has to be ≤0. Consider the case when \lambda -\frac{\sqrt{39}+1}{2}\geq 0 and \lambda -\frac{1-\sqrt{39}}{2}\leq 0.

$$\lambda -\frac{\sqrt{39}+1}{2}\geq 0 \lambda -\frac{1-\sqrt{39}}{2}\leq 0$$

This is false for any \lambda .

$$\lambda \in \emptyset$$

Consider the case when \lambda -\frac{\sqrt{39}+1}{2}\leq 0 and \lambda -\frac{1-\sqrt{39}}{2}\geq 0.

$$\lambda -\frac{1-\sqrt{39}}{2}\geq 0 \lambda -\frac{\sqrt{39}+1}{2}\leq 0$$

The solution satisfying both inequalities is \lambda \in \left[\frac{1-\sqrt{39}}{2},\frac{\sqrt{39}+1}{2}\right].

$$\lambda \in \begin{bmatrix}\frac{1-\sqrt{39}}{2},\frac{\sqrt{39}+1}{2}\end{bmatrix}$$

The final solution is the union of the obtained solutions.

$$\lambda \in \begin{bmatrix}\frac{1-\sqrt{39}}{2},\frac{\sqrt{39}+1}{2}\end{bmatrix}$$