Class 10 Mathematics Ch 4 Short Notes: Quadratic Equations

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1 Definitions

2 Is it quadratic?

3 Forming the equation from a situation

4 Method 1: factorisation

5 Method 2: the quadratic formula

6 Nature of roots

Use this sheet for last-minute revision of Chapter 4, Quadratic Equations. It has every definition, the quadratic formula, the table for the nature of roots and one worked example for each method.

1 Definitions

Learn by heartQuadratic equation

An equation of the form $ax^2+bx+c=0$ , where $a,b,c$ are real numbers and $a\neq0$ . Written with the $x^2$ term first, then the $x$ term, then the number (descending order of degree), this is the standard form.

Learn by heartRoot

A real number $\alpha$ is a root of $ax^2+bx+c=0$ if $a\alpha^2+b\alpha+c=0$ . The roots of the equation are the zeroes of the polynomial $ax^2+bx+c$ (the values of $x$ that make it 0). A quadratic equation has at most two roots.

Learn by heartDiscriminant

The quantity $b^2-4ac$ , written $D$ for short. It decides whether $ax^2+bx+c=0$ has real roots or not.

2 Is it quadratic?

Simplify both sides first, then check for an $x^2$ term and no higher power.
$x(x+1)+8=(x+2)(x-2)$ becomes $x+12=0$ : not quadratic.
$(x+2)^3=x^3-4$ becomes $x^2+2x+2=0$ : is quadratic.

3 Forming the equation from a situation

Situation	Let $x$ be	Equation
Area: length $=2\times$ breadth $+1$ , area 300 m²	the breadth	$x(2x+1)=300$ , i.e. $2x^2+x-300=0$
Two consecutive integers, product $P$	the smaller integer	$x(x+1)=P$
Two numbers with sum $S$ , product $P$	one of the numbers	$x(S-x)=P$
Speed: a trip of $d$ km takes $t$ hours more when the speed is $s$ km/h less	the usual speed	$\dfrac{d}{x-s}-\dfrac{d}{x}=t$

4 Method 1: factorisation

Step 1Find two numbers with product

ac

and sum

b

↓

Step 2Split the middle term

bx

using them, and take out common factors from each pair of terms

↓

Step 3Put each of the two factors equal to zero

Worked example

Q. A hall of area 300 m² has length 1 m more than twice its breadth $x$ , so $2x^2+x-300=0$ . Solve it.

$ac=-600$ , $b=1$ : use $25$ and $-24$ . $2x^2-24x+25x-300=2x(x-12)+25(x-12)=(x-12)(2x+25)=0$ . So $x=12$ or $-12.5$ . Breadth cannot be negative: breadth 12 m, length 25 m.

5 Method 2: the quadratic formula

Formula

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},\quad \text{provided } b^2-4ac\ge0

Worked example

Q. Solve $x^2+7x-60=0$ .

$D=49+240=289$ , $\sqrt{289}=17$ . $x=\dfrac{-7\pm17}{2}$ , so $x=5$ or $x=-12$ .

Note

Completing the square is not taught as a method in the current NCERT book, which states the quadratic formula without deriving it. Use the formula; you do not need to derive it.

6 Nature of roots

Discriminant	Nature of roots
$b^2-4ac\gt0$	two distinct real roots
$b^2-4ac=0$	two equal (coincident) real roots, each $-\dfrac{b}{2a}$
$b^2-4ac\lt0$	no real roots

Worked example

Q. Find the nature of the roots of $2x^2-4x+3=0$ .

$D=16-24=-8\lt0$ : no real roots.

Worked example: equal roots

Q. Find $k$ so that $kx^2-6x+1=0$ has two equal roots.

$D=36-4k=0$ , so $k=9$ . (If you ever get a value of $k$ that makes the number in front of $x^2$ zero, reject it: the equation would no longer be quadratic.)

Remember

“Is it possible?” means: find $D$ . $D\lt0$ not possible. $D\ge0$ gives real roots; then solve and check that a root fits the situation (positive length, age or count).
Reject a negative length, age, speed or count, and write why.
Pole on the boundary of a circular park (NCERT Example 8): a diameter makes a $90^\circ$ angle at any point on the circle, so use Pythagoras theorem to form the equation.

More for this chapter

Notes Important Questions MCQ / Objective NCERT Solutions