Quadratic Equations - Introduction

Check whether $(x-2)^2 + 1 = 2x - 3$ is a quadratic equation.

LHS: $(x-2)^2 + 1 = x^2 - 4x + 4 + 1 = x^2 - 4x + 5$. Given equation: $x^2 - 4x + 5 = 2x - 3$. Rearranging: $x^2 - 4x - 2x + 5 + 3 = 0 \implies x^2 - 6x + 8 = 0$.

Since the resulting equation is in the form $ax^2 + bx + c = 0$ where $a=1, b=-6, c=8$ and $a \neq 0$, it is a quadratic equation.

Represent the following situation mathematically: The area of a rectangular plot is $528\text{ m}^2$. The length of the plot is one more than twice its breadth.

Let the breadth of the plot be $x$ metres. Then, the length is $(2x + 1)$ metres. Area = $\text{Length} \times \text{Breadth} = (2x + 1)x$. We are given the area is $528$. So, $x(2x + 1) = 528 \implies 2x^2 + x - 528 = 0$.

The breadth of the plot satisfies the quadratic equation $2x^2 + x - 528 = 0$.

Calculate the discriminant $D$ for the equation $2x^2 - 5x + 3 = 0$.

Comparing with $ax^2 + bx + c = 0$: $a = 2, b = -5, c = 3$. $D = b^2 - 4ac = (-5)^2 - 4(2)(3) = 25 - 24$. Vertical calculation: $\begin{array}{r} 25 \\ -24 \\ \hline 1 \end{array}$ Thus, $D = 1$.

The discriminant is used to determine the nature of the roots. Since $D > 0$, the equation has two distinct real roots.