fully Factorise \( 2 x^{2}-4 x-1 \)

Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(2x^{2}-4x-1=0\) - step1: Solve using the quadratic formula: \(x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 2\left(-1\right)}}{2\times 2}\) - step2: Simplify the expression: \(x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\times 2\left(-1\right)}}{4}\) - step3: Simplify the expression: \(x=\frac{4\pm \sqrt{24}}{4}\) - step4: Simplify the expression: \(x=\frac{4\pm 2\sqrt{6}}{4}\) - step5: Separate into possible cases: \(\begin{align}&x=\frac{4+2\sqrt{6}}{4}\\&x=\frac{4-2\sqrt{6}}{4}\end{align}\) - step6: Simplify the expression: \(\begin{align}&x=\frac{2+\sqrt{6}}{2}\\&x=\frac{4-2\sqrt{6}}{4}\end{align}\) - step7: Simplify the expression: \(\begin{align}&x=\frac{2+\sqrt{6}}{2}\\&x=\frac{2-\sqrt{6}}{2}\end{align}\) - step8: Rewrite: \(x_{1}=\frac{2-\sqrt{6}}{2},x_{2}=\frac{2+\sqrt{6}}{2}\) To fully factor the expression \( 2x^{2}-4x-1 \), we first need to find its roots by solving the equation \( 2x^{2}-4x-1=0 \). ### Step 1: Solve the quadratic equation Using the quadratic formula, we find the roots: \[ x_{1} = \frac{2 - \sqrt{6}}{2}, \quad x_{2} = \frac{2 + \sqrt{6}}{2} \] ### Step 2: Rewrite the quadratic expression The roots \( x_{1} \) and \( x_{2} \) can be used to express the quadratic in its factored form. The general form of a quadratic can be expressed as: \[ a(x - x_{1})(x - x_{2}) \] where \( a \) is the leading coefficient (which is 2 in this case). ### Step 3: Substitute the roots into the factored form Substituting the roots into the factored form gives us: \[ 2\left(x - \frac{2 - \sqrt{6}}{2}\right)\left(x - \frac{2 + \sqrt{6}}{2}\right) \] ### Step 4: Simplify the factors This can be simplified further: \[ 2\left(x - 1 + \frac{\sqrt{6}}{2}\right)\left(x - 1 - \frac{\sqrt{6}}{2}\right) \] ### Final Factored Form Thus, the fully factored form of \( 2x^{2}-4x-1 \) is: \[ 2\left(x - 1 + \frac{\sqrt{6}}{2}\right)\left(x - 1 - \frac{\sqrt{6}}{2}\right) \] This represents the complete factorization of the quadratic expression.