10th Class Mathematics Quadratic Equations

Quadratic Equations

• 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.

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.