\displaystyle \textbf{Quadratic Equation}

\displaystyle \text{A quadratic equation in one variable }x\text{ is an equation of the form}

\displaystyle ax^2+bx+c=0,\qquad a\ne0,

\displaystyle \text{where }a,b,c\text{ are real numbers and }x\text{ is the unknown variable.}

\displaystyle \text{The condition }a\ne0\text{ is essential. If }a=0,\text{ the }x^2\text{ term}
\displaystyle \text{disappears and the equation is no longer quadratic.}

\displaystyle \text{If }a=0\text{ and }b\ne0,\text{ the equation becomes }bx+c=0,
\displaystyle \text{which is a linear equation.}

\displaystyle \text{In }ax^2+bx+c=0,\text{ the numbers }a,b,c\text{ are called coefficients.}

\displaystyle a\text{ is the coefficient of }x^2\text{ or the quadratic coefficient,}
\displaystyle b\text{ is the coefficient of }x\text{ or the linear coefficient, and}
\displaystyle c\text{ is the constant term.}

\displaystyle \text{The degree of a quadratic equation is }2.

\displaystyle \textbf{Examples of Quadratic Equations}

\displaystyle 21x^2-8x-4=0

\displaystyle \text{Here, }a=21,\qquad b=-8,\qquad c=-4.

\displaystyle 6x^2+5x-6=0

\displaystyle \text{Here, }a=6,\qquad b=5,\qquad c=-6.

\displaystyle x^2-9=0

\displaystyle \text{Here, }a=1,\qquad b=0,\qquad c=-9.

\displaystyle \text{Thus, either }b\text{ or }c\text{ may be zero in a quadratic equation,}
\displaystyle \text{but the quadratic coefficient }a\text{ must always be non-zero.}

\displaystyle \textbf{Roots of a Quadratic Equation}

\displaystyle \text{A value of }x\text{ that satisfies a quadratic equation is called a root}
\displaystyle \text{or solution of the quadratic equation.}

\displaystyle \text{A quadratic equation has two roots when the roots are counted according to}
\displaystyle \text{their multiplicity. The roots may be distinct or equal, and they may be real}
\displaystyle \text{or non-real, depending on the discriminant.}

\displaystyle \text{If }p\text{ and }q\text{ are the roots of }ax^2+bx+c=0,\text{ then}

\displaystyle ax^2+bx+c=a(x-p)(x-q).

\displaystyle \text{Therefore, the quadratic equation can be written as}

\displaystyle a(x-p)(x-q)=0.

\displaystyle \text{Since }a\ne0,\text{ we have}

\displaystyle (x-p)(x-q)=0.

\displaystyle \therefore x=p\qquad\text{or}\qquad x=q.

\displaystyle \text{If }p=q,\text{ the equation has two equal roots and can be written as}

\displaystyle a(x-p)^2=0.

\displaystyle \textbf{Solving Quadratic Equations}

\displaystyle \text{The following methods are commonly used to solve quadratic equations:}

\displaystyle \text{(i) Factorization Method}
\displaystyle \text{(ii) Quadratic Formula}
\displaystyle \text{(iii) Completing the Square Method}

\displaystyle \textbf{1. Factorization Method}

\displaystyle \text{Step 1: Write the quadratic equation in the standard form}
\displaystyle ax^2+bx+c=0.

\displaystyle \text{Step 2: Factorize the quadratic expression completely into linear factors.}

\displaystyle ax^2+bx+c=a(x-p)(x-q).

\displaystyle \text{Step 3: Use the zero-product property.}

\displaystyle a(x-p)(x-q)=0

\displaystyle \Rightarrow x-p=0\qquad\text{or}\qquad x-q=0.

\displaystyle \therefore x=p\qquad\text{or}\qquad x=q.

\displaystyle \textbf{Example: }\quad x^2-5x+6=0

\displaystyle x^2-5x+6=0

\displaystyle \Rightarrow (x-2)(x-3)=0

\displaystyle \Rightarrow x-2=0\qquad\text{or}\qquad x-3=0

\displaystyle \therefore x=2\qquad\text{or}\qquad x=3.

\displaystyle \text{Hence, the roots are }2\text{ and }3.

\displaystyle \textbf{2. Quadratic Formula}

\displaystyle \text{For the quadratic equation }ax^2+bx+c=0,\quad a\ne0,

\displaystyle \text{first identify the values of }a,\ b\text{ and }c.

\displaystyle \text{The roots are given by the quadratic formula}

\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.

\displaystyle \text{The symbol }\pm\text{ means that two values are to be considered:}

\displaystyle x=\frac{-b+\sqrt{b^2-4ac}}{2a} \qquad\text{and}\qquad x=\frac{-b-\sqrt{b^2-4ac}}{2a}.

\displaystyle \textbf{Discriminant and Nature of Roots}

\displaystyle \text{The expression }b^2-4ac\text{ is called the discriminant of the quadratic}
\displaystyle \text{equation and is usually denoted by }D.

\displaystyle D=b^2-4ac.

\displaystyle \text{The value of }D\text{ determines the nature of the roots.}

\displaystyle \text{(i) If }D>0,\text{ the equation has two distinct real roots.}

\displaystyle \text{(ii) If }D=0,\text{ the equation has two equal real roots.}

\displaystyle \text{In this case, }x=-\frac{b}{2a}.

\displaystyle \text{(iii) If }D<0,\text{ the equation has no real roots.}

\displaystyle \text{If complex numbers are considered, it has two non-real complex roots.}

\displaystyle \textbf{3. Completing the Square Method}

\displaystyle \text{A quadratic equation can also be solved by rewriting its quadratic expression}
\displaystyle \text{as a perfect square. This method is called completing the square.}

\displaystyle \text{For example, consider }x^2+6x+5=0.

\displaystyle x^2+6x=-5

\displaystyle \Rightarrow x^2+6x+9=4

\displaystyle \Rightarrow (x+3)^2=4

\displaystyle \Rightarrow x+3=\pm2

\displaystyle \therefore x=-1\qquad\text{or}\qquad x=-5.

Discover more from ICSE / ISC / CBSE Mathematics Portal for K12 Students

Subscribe to get the latest posts sent to your email.