Question

solve.
0 = x² - 4x - 20
x = {-2 ± 6√2}
x = {2 ± 6√2}
x = {2 ± 2√6}
x = {-2 ± 2√6}

Explanation:

Step1: Identify the quadratic equation

The equation is \(x^2 - 4x - 20 = 0\). For a quadratic equation \(ax^2 + bx + c = 0\), the quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Here, \(a = 1\), \(b=-4\), \(c = -20\).

Step2: Calculate the discriminant

The discriminant \(D=b^2 - 4ac\). Substitute \(a = 1\), \(b=-4\), \(c=-20\):
\(D=(-4)^2-4\times1\times(-20)=16 + 80 = 96\).

Step3: Simplify the square root of the discriminant

\(\sqrt{96}=\sqrt{16\times6}=4\sqrt{6}\). Wait, no, wait, let's recalculate. Wait, \(b=-4\), so \(-b = 4\). Then \(x=\frac{4\pm\sqrt{16+80}}{2}=\frac{4\pm\sqrt{96}}{2}\). Simplify \(\sqrt{96}=\sqrt{16\times6}=4\sqrt{6}\)? Wait, no, \(96 = 16\times6\), so \(\sqrt{96}=4\sqrt{6}\)? Wait, no, wait \(96 = 16\times6\), so \(\sqrt{96}=4\sqrt{6}\), then \(\frac{4\pm4\sqrt{6}}{2}=2\pm2\sqrt{6}\)? Wait, no, wait the options have \(2\pm2\sqrt{6}\)? Wait, the third option is \(x=\{2\pm2\sqrt{6}\}\). Wait, let's re - check the discriminant:

Wait, equation is \(x^2-4x - 20=0\). \(a = 1\), \(b=-4\), \(c=-20\). \(b^2-4ac=(-4)^2-4(1)(-20)=16 + 80 = 96\). \(\sqrt{96}=\sqrt{16\times6}=4\sqrt{6}\). Then \(x=\frac{4\pm4\sqrt{6}}{2}=2\pm2\sqrt{6}\). So the solution is \(x = \{2\pm2\sqrt{6}\}\), which is the third option (the one with \(x=\{2\pm2\sqrt{6}\}\)).

Answer:

\(x=\{2\pm2\sqrt{6}\}\) (the third option in the given choices)