ise 9 Let \( h(x)=x^{2}-4 x-c \). a) Find a nonzero value for c such that \( h(c)=c \). b) Find and simplify \( h(x-2) \) and \( h(x+a) \). Use the c-value found in part (a). Solution (b) \( h(x-2) \) h(x)=x \( 2-4 x-6 \) \[ h(x)=x^{2}-4 x-c \] \[ h(x+a) \] \[ (x-2)^{2}=(x-2) x \] such that \( \overparen{h(c)}=C \) \[ \begin{array}{l} h(x+a) \\ h(x-2)=(x-2)^{2}-4(x-2)-6 \end{array} \] \[ x \rightarrow x-2 \] \( h(x-2)=x^{2}-2 \cdot x \cdot 2+2^{2}-4 x+8-6 \) apply \( (a-b)^{2}=0 \) quachatic eq. \( \begin{array}{c}\text { Applying the } \theta \text { - } \\ \text { Property }\end{array} \) \( \Rightarrow c=0 \) or \( c-6=0 \) \( c=6 \) \( =0 \) Factoring Method Applying the 0 -factor \( \ln (x-2)=x^{2}-8 x+6 \) \[ \overbrace{\substack{ \\h(x+a) \\ x \rightarrow x+a}}^{\ln (x-2)=x-0 x+\infty}(x+a)^{2}-4(a+8)-6 \]

Let’s dive into the fascinating history of quadratics! The quadratic equation, particularly of the form \(ax^2 + bx + c = 0\), has been around since ancient times, with mathematicians like Babylonian scholars using geometric methods to solve specific cases over 4,000 years ago. Fast forward to the 17th century, and we see giants like Descartes and Fermat making strides that established the foundations of modern algebra. Isn’t it amazing how these equations have shaped mathematics through the ages? Now, let’s talk about real-world applications! Quadratic equations pop up in various fields, including physics, engineering, and finance. For instance, they are essential in calculating the trajectory of objects in motion—think of throwing a ball or launching a rocket! Additionally, businesses use quadratics to model profit maximization and cost analysis. Understanding how quadratics function can provide invaluable insights into getting things just right in the real world!