Question

equality notation, interval notation, and as a shaded graph 2) $x^{2}-2x - 5 < 0$

Explanation:

Step1: Find the roots of the quadratic equation

For the quadratic equation $x^{2}-2x - 5=0$, use the quadratic formula $x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$, where $a = 1$, $b=-2$, and $c=-5$. Then $x=\frac{2\pm\sqrt{(-2)^{2}-4\times1\times(-5)}}{2\times1}=\frac{2\pm\sqrt{4 + 20}}{2}=\frac{2\pm\sqrt{24}}{2}=\frac{2\pm2\sqrt{6}}{2}=1\pm\sqrt{6}$. The roots are $x_1=1+\sqrt{6}\approx1 + 2.45=3.45$ and $x_2=1-\sqrt{6}\approx1-2.45=-1.45$.

Step2: Determine the solution of the inequality

Since the quadratic function $y = x^{2}-2x - 5$ is a parabola opening upwards (because $a = 1>0$), the solution of the inequality $x^{2}-2x - 5<0$ is the set of $x$ - values between the two roots.

• Inequality notation: $1-\sqrt{6}
• Interval notation: $(1-\sqrt{6},1+\sqrt{6})$.
• Shaded graph: On a number - line, mark the points $x = 1-\sqrt{6}\approx - 1.45$ and $x = 1+\sqrt{6}\approx3.45$ with open circles (because the inequality is strict, $<$) and shade the region between these two points.

Answer:

Inequality notation: $1-\sqrt{6}Interval notation: $(1 - \sqrt{6},1+\sqrt{6})$
Shaded graph: Mark $x\approx - 1.45$ and $x\approx3.45$ with open - circles and shade the region between them on the number - line.