Question

exit ticket
use the quadratic formula to solve ( x^2 - 4x + 1 = 0 ) for ( x ). round the solutions to the nearest hundredth.

Explanation:

Step1: Identify a, b, c

For quadratic equation \(ax^2 + bx + c = 0\), here \(a = 1\), \(b = -4\), \(c = 1\).

Step2: Quadratic formula

Quadratic formula is \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\). Substitute values: \(x=\frac{-(-4)\pm\sqrt{(-4)^2 - 4\times1\times1}}{2\times1}=\frac{4\pm\sqrt{16 - 4}}{2}=\frac{4\pm\sqrt{12}}{2}\).

Step3: Simplify \(\sqrt{12}\)

\(\sqrt{12}=2\sqrt{3}\approx3.464\). So \(x=\frac{4\pm3.464}{2}\).

Step4: Calculate two solutions

First solution: \(x=\frac{4 + 3.464}{2}=\frac{7.464}{2}=3.732\approx3.73\). Second solution: \(x=\frac{4 - 3.464}{2}=\frac{0.536}{2}=0.268\approx0.27\).

Answer:

\(x\approx3.73\) and \(x\approx0.27\)