CBSE Class 10 Mathematics Quadratic Equations Notes Set 05

Nature of Roots

Let \( \alpha \) and \( \beta \) be the roots of the quadratic equation \( ax^2 + bx + c = 0,\ a \neq 0 \), then\[ \alpha = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \quad \text{and} \quad \beta = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \]

Mind it. If \( a, b, c \in Q \) and \( p + \sqrt{q} \) is one root of quadratic equation then the other root must be the conjugate i.e., \( p - \sqrt{q} \) and vice-versa. Here p and q are real numbers.

Example 1. Without solving for 'x', examine the nature of roots of the equations: (i) \( 2x^2 + 2x + 3 = 0 \); (ii) \( x^2 + 5x + 6 = 0 \); (iii) \( x^2 - 5x - 2 = 0 \).
Answer:
(i) Comparing \( 2x^2 + 2x + 3 = 0 \) with \( ax^2 + bx + c = 0 \), we get \( a = 2,\ b = 2 \) and \( c = 3 \). \( D = b^2 - 4ac = (2)^2 - 4 \times 2 \times 3 = -20 \), which is negative. \( \therefore \) The given equation has no real root.
(ii) Comparing \( x^2 + 5x + 6 = 0 \) with \( ax^2 + bx + c = 0 \), we get \( a = 1,\ b = 5 \) and \( c = 6 \). \( D = b^2 - 4ac = (5)^2 - 4(1)(6) = 25 - 24 = 1 \). As \( D > 0 \), given equation has distinct and real roots.
(iii) Comparing \( x^2 - 5x - 2 = 0 \) with \( ax^2 + bx + c = 0 \), we get \( a = 1,\ b = -5 \) and \( c = -2 \). \( D = b^2 - 4ac = (-5)^2 - 4 \times 1 \times (-2) = 25 + 8 = 33 \), which is positive but not a perfect square. \( \therefore \) The roots of the given equation are distinct and unequal.

Example 2. Find the value of p, for which the roots of the equation \( (3p + 1)x^2 + (11 + p)x + 9 = 0 \) are equal.
Answer:
Compare the given equation with \( ax^2 + bx + c = 0 \); we get \( a = 3p + 1,\ b = 11 + p \) and \( c = 9 \).
\( D = b^2 - 4ac = (11 + p)^2 - 4(3p + 1) \times 9 = 121 + 22p + p^2 - 108p - 36 = p^2 - 86p + 85 \).
By factorising using middle term splitting, \( p^2 - 85p - p + 85 = (p - 85)(p - 1) \).
Since the roots are equal, \( D = 0 \Rightarrow (p - 85)(p - 1) = 0 \Rightarrow p = 85 \) or \( p = 1 \).

Formation of Quadratic Equations

Suppose \( \alpha \) and \( \beta \) are the two roots of the quadratic equation \( ax^2 + bx + c = 0 \), then the quadratic equation can be written as\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \]i.e., \( x^2 - (\alpha + \beta)x + (\alpha\beta) = 0 \). Let 2 and 3 be the roots of some quadratic equation, then \( x^2 - 5x + 6 = 0 \) is the corresponding quadratic equation.

Example 3. Construct the quadratic equation whose roots are given below: (i) 3, \( -3 \); (ii) \( 3 + \sqrt{3},\ 3 - \sqrt{3} \).
Answer:
(i) The given roots are 3 and \( -3 \). Sum of the roots \( = 3 + (-3) = 0 \) and the product of the roots \( = 3(-3) = -9 \). \( \therefore \) The required quadratic equation is \( x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \Rightarrow x^2 - (0)x + (-9) = 0 \), i.e., \( x^2 - 9 = 0 \).
(ii) The given roots are \( 3 + \sqrt{3} \) and \( 3 - \sqrt{3} \). Sum of the roots \( = 3 + \sqrt{3} + 3 - \sqrt{3} = 6 \) and the product of the roots \( = (3 + \sqrt{3})(3 - \sqrt{3}) = 9 - 3 = 6 \). \( \therefore \) The required quadratic equation is \( x^2 - 6x + 6 = 0 \).

Example 4. If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( ax^2 + bx + c = 0,\ (a \neq 0) \) then find the values of: (i) \( \alpha^2 + \beta^2 \); (ii) \( \alpha^3 + \beta^3 \).
Answer:
We know, sum of roots \( (\alpha + \beta) = -\frac{b}{a} \), and product of roots \( (\alpha\beta) = \frac{c}{a} \); therefore:
(i) \( (\alpha + \beta)^2 = \alpha^2 + \beta^2 + 2\alpha\beta \Rightarrow \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(-\frac{b}{a}\right)^2 - \frac{2c}{a} = \frac{b^2 - 2ac}{a^2} \).
(ii) \( \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 + \beta^2 - \alpha\beta) = \left(-\frac{b}{a}\right)\left(\frac{b^2 - 2ac}{a^2} - \frac{c}{a}\right) = \left(-\frac{b}{a}\right)\left(\frac{b^2 - 2ac - ac}{a^2}\right) = -\frac{b(b^2 - 3ac)}{a^3} \).

Formation of a New Quadratic Equation by Changing the Roots of a Given Quadratic Equation

Let \( \alpha, \beta \) be the roots of a quadratic equation \( ax^2 + bx + c = 0 \), then we can form a new quadratic equation as per the following rules.
1. A quadratic equation whose roots are p more than the roots of the equation \( ax^2 + bx + c = 0 \) (i.e., the roots are \( \alpha + p \) and \( \beta + p \)). The required quadratic equation will be \( a(x - p)^2 + b(x - p) + c = 0 \).
2. A quadratic equation whose roots are less by p than the roots of the equation \( ax^2 + bx + c = 0 \) (i.e., the roots are \( \alpha - p \) and \( \beta - p \)). The required equation will be \( a(x + p)^2 + b(x + p) + c = 0 \).
3. The equation whose roots are p times the roots of the given equation can be written as \( a\left(\frac{x}{p}\right)^2 + b\left(\frac{x}{p}\right) + c = 0 \).
4. The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) can be written as \( a\left(\frac{1}{x}\right)^2 + b\left(\frac{1}{x}\right) + c = 0 \) i.e., \( cx^2 + bx + a = 0 \).
5. A quadratic equation whose roots are \( \frac{1}{p} \) times the roots of the equation \( ax^2 + bx + c = 0 \) (i.e., the roots are \( \frac{\alpha}{p} \) and \( \frac{\beta}{p} \)). The required equation is \( a(px)^2 + b(px) + c = 0 \).
6. A quadratic equation whose roots are the negative of the roots of the equation \( ax^2 + bx + c = 0 \) (i.e., the roots are \( -\alpha \) and \( -\beta \)). The required equation is \( ax^2 - bx + c = 0 \).
7. A quadratic equation whose roots are the square of the roots of the equation \( ax^2 + bx + c = 0 \) (i.e., the roots are \( \alpha^2 \) and \( \beta^2 \)). The required equation will be \( a^2x^2 - (b^2 - 2ac)x + c^2 = 0 \).

Summary

An equation of the form \( ax^2 + bx + c = 0,\ a \neq 0 \) is called a quadratic equation in x.

Roots of a quadratic equation. The value of x which satisfies the quadratic equation i.e., \( ax^2 + bx + c = 0 \) are called roots of a quadratic equation.

Pure quadratic equations \( ax^2 + c = 0 \) can be solved by taking square root.

Solution of adfected quadratic equations by: 1. Factorisation, 2. Completing the square, 3. Quadratic formula.