Category : 10th Class

• Quadratic equation: An equation of the form $a{{x}^{2}}+bx+c=0$ where a, b and$c\in R$and $a\ne 0$is called a quadratic equation.

If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.

Note: (i) An equation of degree 2 is called a quadratic equation.

(ii) The quadratic equation of the form $a{{x}^{2}}+bx+c=0$.

Solution or roots of a quadratic equation: If p(x) = 0 is a quadratic equation, then the zeros of the polynomial p(x) are called the solutions or roots of the quadratic equation p(x)=0.

Note: (i) Since the degree of a quadratic equation is 2, it has 2 roots or solutions.

(ii) x = a is the root of p(x) = 0, if p(a) = 0.

(iii) Finding the roots of a quadratic equation is called solving the quadratic equation.

• Methods of solving a quadratic equation: There are different methods of solving a quadratic equation.

(a) Factorization method

(i) Splitting the middle term

(ii) Completing the square

(b) Formula method

(a) (i) Splitting the middle term: Consider the quadratic equation$a{{x}^{2}}+bx+c=0$.

Step 1: Find the product of the coefficient of${{x}^{2}}$and the constant term i.e., ac.

Step 2: (a) If ac is positive, then choose two factors of ac, whose sum is equal to b (the coefficient of the middle term).

(b)  If ac is negative, then choose two factors of ac, whose difference is equal to b (the coefficient of the middle term).

Step 3: Express the middle terms as the sum (or difference) of the two factors obtained in step 2. [Now the quadratic equation has 4 terms]

• Step 4: Separate the term common to the first two terms and then write the first two terms as a product. Take the common term (binomial) out of the last two terms and get another factor so that the last two terms are written as a product.

Step 5: Express the given quadratic equation as a product of two binomials, and solve them.

Step 6: The two values obtained in step 5 are the roots of the given quadratic equation.

Note: An important property used in solving a quadratic equation by splitting the middle term.

"If ab = 0, then either a = 0, or b = 0 or both a and b are 0, where 'a' and 'b' are real numbers"

(ii) Completing the square: In some cases where the given quadratic equation can be solved by factorization, a suitable term is added and subtracted. Then terms are regrouped in such a manner that a square is completed by three of the terms. The equation is then solved using factorization method.

Note: Usually, the term added and subtracted is the square of half the coefficient of x.

1. Formula method: The roots of a quadratic equation$a{{x}^{2}}+bx+c=0$are given by $\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$provided${{b}^{2}}-4ac\ge 0$. This formula for finding the roots of a quadratic equation is called the quadratic formula.

Note: The roots of the quadratic equation using the quadratic formula are

$x=\frac{-b\sqrt{{{b}^{2}}-4ac}}{2a}$and$x=\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$

Nature of roots: For a quadratic equation $a{{x}^{2}}+bx+c=0,$${{b}^{2}}-4ac$(denoted by D) is called the discriminate.

 Value of Discriminate Roots Nature of roots No. of Roost Greater than (positive) $\frac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$and $\frac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$ Real and distinct 2 Lesser than 0 (Negative) $\frac{-b+i\sqrt{|D|}}{2a}$and $\frac{-b-i\sqrt{|D|}}{2a}$ Complex or imaginary 0 Equal to 0 $\frac{-b}{2a},\frac{-b}{2a}$ Real and 2 coincident roots or repeated roots

If ${{\text{b}}^{\text{2}}}-\text{4ac}>0$, then $\sqrt{{{b}^{2}}-4ac}$is a real number and the quadratic equation $a{{x}^{2}}+bx+c=0$has two real roots a and p.

If ${{\text{b}}^{\text{2}}}-\text{4ac}<0$, then $\sqrt{{{\text{b}}^{\text{2}}}-\text{4ac}}$ is not a real number and the quadratic equation $a{{x}^{2}}+bx+c=0$ has no real root.

• Quadratic equations can be applied to solve word problems.

Note: Any root of the quadratic equation that does not satisfy the condition of the problem is discarded.