Class 10 Maths · 2026–27 study draft

Quadratic Equations: Understand, Solve and Check

A quadratic equation describes a relationship in which the highest power of the unknown is two. You might meet one when a rectangle's dimensions depend on the same number, or when the product of two consecutive numbers is known. Start by rearranging the equation, then choose a method and check each answer in the original situation.

What you will learn

Recognise standard form and identify the coefficients with their signs.
Solve by factorisation and the quadratic formula.
Use the discriminant to predict real roots.
Translate a context into an equation and reject invalid contextual answers.

The CBSE 2026–27 syllabus includes standard form, real roots by factorisation and the quadratic formula, the discriminant and nature of roots, and everyday situational problems. These notes focus on that scope; complex roots are not required here.

Core concepts

Put every term on one side

Write $ax^2+bx+c=0$ , where $a\ne0$ . Missing terms have coefficient zero. For example, $2x^2=7x-3$ becomes $2x^2-7x+3=0$ , so $b=-7$ , not $7$ .

Factorisation uses the zero-product rule

If a product is zero, at least one factor is zero. After writing $(x-r)(x-s)=0$ , solve $x-r=0$ or $x-s=0$ . This rule does not apply to a sum or to a product equal to a non-zero number.

The discriminant tells you what to expect

Calculate $D=b^2-4ac$ . A positive value gives two distinct real roots, zero gives two equal real roots, and a negative value gives no real roots. An irrational square root can be left in exact surd form.

Important formulas

ax^2+bx+c=0\quad(a\ne0)

D=b^2-4ac

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\quad(D\ge0)

Three worked examples

Example 1

Solve $x^2-7x+10=0$ by factorisation.

Find two numbers whose product is $10$ and sum is $-7$ : they are $-5$ and $-2$ .
Write $(x-5)(x-2)=0$ . Set each factor equal to zero.
Check: $25-35+10=0$ and $4-14+10=0$ .

Answer: $x=2$ or $x=5$ .

Example 2

Solve $2x^2+3x-7=0$ using the formula.

Here $a=2$ , $b=3$ , $c=-7$ . Hence $D=3^2-4(2)(-7)=65>0$ .
Substitute with the signs intact: $x=\frac{-3\pm\sqrt{65}}{4}$ .
Both roots are real and distinct. Their sum is $-3/2$ and product is $-7/2$ , consistent with the coefficients.

Answer: $x=\frac{-3+\sqrt{65}}4$ or $x=\frac{-3-\sqrt{65}}4$ .

Example 3

A rectangular reading area has width $x$ metres and length $(x+3)$ metres. Its area is $40$ square metres. Find the dimensions.

Area equals width times length: $x(x+3)=40$ , or $x^2+3x-40=0$ .
Factorise: $(x+8)(x-5)=0$ , so $x=-8$ or $x=5$ .
A physical width must be positive, so discard $-8$ . With $x=5$ , length is $8$ and $5\times8=40$ .

Answer: Width $5$ m; length $8$ m.

Common errors to avoid

Do not lose the minus sign when identifying a coefficient after rearranging.
The whole numerator of the quadratic formula is divided by $2a$ , not only the square root.
Keep both algebraic roots until the context tells you whether one is invalid.
A negative discriminant means no real roots, not that the equation is not quadratic.

Try it yourself

Attempt these original questions before checking the answer and reasoning.

Solve $x^2-9x+20=0$ .
Check answer and reasoning
$x=4$ or $x=5$ .
$(x-4)(x-5)=0$ ; substitution verifies both roots.
State the nature of the roots of $9x^2-12x+4=0$ .
Check answer and reasoning
Two equal real roots, both $2/3$ .
$D=144-144=0$ and $(3x-2)^2=0$ .
Does $x^2+2x+5=0$ have real roots?
Check answer and reasoning
No real roots.
$D=4-20=-16<0$ .

Related chapter guides

Introduction to Trigonometry · Arithmetic Progressions

Both algebra guides turn relationships into equations; compare a quadratic unknown with the linear rule for an AP term.

Explore all Class 10 Maths chapters

Official syllabus and draft review

Aligned to the official CBSE Class X Mathematics syllabus for 2026–27 (041 and 241). Check the CBSE curriculum portal for updates and your school's guidance.

Original Mathslume study drafts, checked against the syllabus with independently calculated answers. Not official CBSE/NCERT solutions or an endorsement. A qualified teacher's editorial sign-off is still recommended before publication.

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