CBSE Class 10 Mathematics Quadratic Equations Notes Set 09

Quadratic Equations: Key Ideas

A quadratic equation has the form \( ax^2 + bx + c = 0 \) with a ≠ 0. It has at most two roots. If b = 0 it is a pure quadratic, if c = 0 then x = 0 is one root.
Factorisation: split the middle term so that the product of the two parts equals ac, then equate each factor to zero.
Completing the square: divide by a, add and subtract the square of half the coefficient of x, and the equation becomes (x + p)² = r.
Quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Discriminant and Nature of Roots

The discriminant is D = b² − 4ac.
D > 0: two distinct real roots. D = 0: two equal real roots, each equal to −b/(2a). D < 0: no real roots (a pair of complex roots).
If D is a perfect square and a, b, c are rational, the roots are rational. If D is positive but not a perfect square, the roots are irrational conjugates.
For equal roots, set D = 0 and solve for the unknown constant, for example 3x² + kx + 3 = 0 gives k² = 36, so k = ±6.

Sum and Product of Roots

For roots α and β: \( \alpha + \beta = -\frac{b}{a} \) and \( \alpha\beta = \frac{c}{a} \).
The equation with given roots is x² − (α + β)x + αβ = 0.
Useful identities: \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), \( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta \) and \( \alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta) \).

Special Conditions

Roots equal in magnitude and opposite in sign: b = 0.
Roots are reciprocals of each other: a = c.
One root is n times the other: \( nb^2 = (n+1)^2 ac \).
One root is zero: c = 0. Both roots are zero: b = c = 0.
Common root of two equations: subtract suitably to eliminate x², find the common value of x, and substitute back.
If one root is a + √b with rational coefficients, the other is its conjugate a − √b.

Equations Reducible to Quadratics

Many equations become quadratic after a substitution: put y = x + 1/x for a reciprocal-type quartic, y = 2x/(x − 5) when a fraction repeats, or t = 7^x for an exponential equation.
Always substitute the answers back, because clearing denominators or squaring can create extra roots, and check the restrictions on x.

Word Problems

Define the unknown, translate the statement into an equation using the relevant formula (time = distance ÷ speed, area, Pythagoras, digits 10a + b), reduce to a quadratic, solve, and reject roots that are negative or impossible.
For a boat in a stream: downstream speed = boat + stream, upstream speed = boat − stream.