4. Quadratic Equations

6 topics1 h

Chapter 4 of NCERT Mathematics for Class 10: Quadratic Equations. Three sections — Recognising and forming one, Roots by factorising and What the discriminant decides. Six videos, 1 h in all.

Worked answers to the 13 exercise questions in this chapter: Exercise 4.1 · Exercise 4.2 · Exercise 4.3

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Part 1

Recognising and forming one

14 minThe shape an equation has to have before it counts as quadraticBeing quadratic is not something you can SEE. It is a property an equation has after it has been tidied, and two of the equations here change their answer the moment you do the arithmetic: one with an x2 on both sides that turns out to be linear, one with a cube on both sides that turns out to be quadratic. Over 1980 equations, 712 of them look quadratic and are not.14 min · 12 parts15 minTurning a described situation into an equation of that shapeA situation turns quadratic for a reason you can point at: the sentence that ties it together MULTIPLIES two quantities that both move when the unknown moves. Over 3969 products of two linear expressions, both factors moving gives degree two every single time, and one fixed factor never does - which is why the same floor measured by its perimeter stays linear.15 min · 14 parts

Part 2

Roots by factorising

11 minWhy a root of the equation is the same thing as a zero of the polynomialA ROOT of the equation and a ZERO of the polynomial sound like two findings about the same number — one substitution checks both, and the equals sign is never touched.11 min · 12 parts16 minSplitting the middle term, then setting each factor to zeroFactorising does not solve a quadratic equation. NOUGHT does. On a grid of ordinary numbers, 20 pairs multiply to 12 and not one contains a nought; 385 pairs multiply to NOTHING and every single one does. That asymmetry is the whole method - and it is a fact about the real numbers, not about multiplying, because on a clock face 17 pairs multiply to nothing with neither of them being nothing.16 min · 14 parts

Part 3

What the discriminant decides

14 minThe formula this book hands you for the roots, and how to apply itThe quadratic formula is usually handed over ready-made. It is four lines of working away, and those four lines hand you two things the bare formula does not mention: where the plus-or-minus comes from, and why b squared minus four a c is not allowed to go negative.14 min · 14 parts15 minWhy the sign of b² − 4ac settles how many real roots existYou can say how many real roots an equation has without solving it, and without even knowing where they are. Both roots start at the same place and are pushed apart by one term, so the only question is what that push is - a real distance, no distance, or no real number to push by.15 min · 15 parts

What you will be able to do

  • State the four conditions the standard form imposes: one variable, real coefficients, highest power exactly two, and a leading coefficient that is not zero
  • Choose a single unknown for a described situation and justify why one letter is enough
  • State what it takes for a number to be a root of a quadratic equation, in the general form the chapter gives with α
  • State the property of zero that licenses equating each factor to zero, and explain why the same move fails when the product equals any other number
  • Write the root formula from memory in the form the chapter prints it, and read each part of it aloud correctly
  • Compute b² − 4ac from an equation in standard form, keeping every sign

What this chapter assumes you already know

  • Polynomials in one variable, their degree, and the quadratic polynomial written in the form ax² + bx + c — Class X Chapter 2
  • Expanding a product of two binomials, and the expansions of (x + k)² and (x + k)³
  • Collecting like terms and moving terms across an equals sign
  • What a real number is, and that real numbers include integers, fractions and surds such as √2
  • The difference between an expression and an equation
  • Expanding a product of two linear expressions in one variable
  • Area of a rectangle, and perimeter of a rectangle
  • The relation between distance, uniform speed and time

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "It has an x², so it is quadratic."
  • "It has an x³, so it cannot be quadratic."
  • "Standard form is just neatness."
  • "a can be any number."
  • "b or c must be present."
  • "The coefficients have to be whole numbers."
  • "Dividing the whole equation by 6 changes the problem."
  • "Two unknown quantities need two letters."