Class 10 Maths Notes Chapter 4 — Quadratic Equations#

Key terms#

  • Quadratic equation — an equation $ax^2 + bx + c = 0$, where $a$, $b$, $c$ are real numbers and $a \neq 0$. This is its standard form.
  • Root — a real number $\alpha$ is a root if $a\alpha^2 + b\alpha + c = 0$.
  • The roots of $ax^2 + bx + c = 0$ are the same as the zeroes of the polynomial $ax^2 + bx + c$.
  • A quadratic equation has at most two roots.
  • To check whether an equation is quadratic, expand and simplify it first. It is quadratic only if the $x^2$ term remains with a non-zero coefficient (and no higher power remains).

Solving by factorisation#

  1. Write the equation in standard form $ax^2 + bx + c = 0$.
  2. Split the middle term: find two numbers whose sum is $b$ and whose product is $ac$.
  3. Factorise by grouping into two linear factors.
  4. Put each factor equal to zero and solve.

Example: $2x^2 - 5x + 3 = 0$. We need two numbers with sum −5 and product $2 \times 3 = 6$. They are −2 and −3.

$$2x^2 - 2x - 3x + 3 = 0$$

$$\Rightarrow 2x(x - 1) - 3(x - 1) = 0$$

$$\Rightarrow (x - 1)(2x - 3) = 0$$

$$\Rightarrow x = 1 \text{ or } x = \frac{3}{2}$$

Quadratic formula#

The roots of $ax^2 + bx + c = 0$ are

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

provided $b^2 - 4ac \geq 0$.

This is also called Sridharacharya’s formula.

Nature of roots#

The discriminant of $ax^2 + bx + c = 0$ is

$$D = b^2 - 4ac$$
Discriminant Nature of roots Roots
$D > 0$ Two distinct real roots $\dfrac{-b + \sqrt{D}}{2a}$, $\dfrac{-b - \sqrt{D}}{2a}$
$D = 0$ Two equal real roots $-\dfrac{b}{2a}$, $-\dfrac{b}{2a}$
$D < 0$ No real roots —
xD > 0two distinct real rootsxD = 0two equal real rootsxD < 0no real roots
The graph of $y = ax^2 + bx + c$ for each sign of $D$ (with $a > 0$). Real roots are where it meets the x-axis.
  • Real roots exist only when $D \geq 0$.
  • For “find $k$ so that the roots are equal”, put $D = 0$ and solve for $k$.

Word problems#

  • Choose a variable, form the equation from the given condition, and write it in standard form.
  • Solve by factorisation or by the formula.
  • Reject roots that do not make sense, such as a negative length, age, speed or number of items.
  • For “is it possible?” questions, find $D$. If $D < 0$, the situation is not possible.

Common mistakes#

  • Using the formula before writing the equation in standard form, so $a$, $b$, $c$ have the wrong signs.
  • Calling an equation quadratic without simplifying. $(x - 2)(x + 1) = (x - 1)(x + 3)$ simplifies to a linear equation.
  • Writing $D = b^2 + 4ac$, or forgetting that $b^2$ is always non-negative even when $b$ is negative.
  • Saying “two equal roots” means one root. A quadratic with $D = 0$ has two equal roots.
  • Keeping both roots in a word problem when one is impossible.

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