Quadratic EquationsQuadratic Equations
Here is the precise definition. A quadratic equation in the variable is an equation of the form
where , and are
Seen this before?
In the Polynomials chapter you hunted for the zeroes of . A quadratic equation asks the same question in a new way — for which is equal to 0? — so the zeroes of the polynomial are exactly the solutions of the equation.
Put another way: whenever is a polynomial of
The whole definition hangs on those three symbols, . Drag and watch both the equation and its graph — what happens the moment reaches 0?
The term vanishes, the curve straightens into a line, and a linear equation is all that is left. In standard form each number has a job: is the coefficient of , is the coefficient of , and is the constant term. So , rewritten as , has
Quadratic equations turn up in all kinds of everyday situations, and in many other parts of mathematics. Let us look at a few.
Marbles at Break Time
John and Jivanti have 45 marbles between them. In a game at break time, each of them loses 5. When they multiply the numbers of marbles they have left, the answer is 124. How many did each of them start with?
Play detective first. Drag John's starting number of marbles: the piles show who has what, and the rectangle of little squares shows the product of what is left. Make the product exactly 124.
John started with
Example 1 : Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹750. We would like to find out the number of toys produced on that day.
Solution :
(i) Let John's starting number of marbles be .
Then Jivanti started with
After losing 5, John has marbles, and Jivanti has
Their product is
We are told that this product is 124, so , i.e. . Multiplying through by −1 gives
So John's starting number of marbles satisfies the quadratic equation — this is the mathematical form of the problem. And you already know two numbers that satisfy it: the slider found them!
The Toy Workshop
Now part (ii). In a small workshop, the more toys are made in a day, the cheaper each one becomes: the cost of each toy, in rupees, is 55 minus the number of toys made that day. One day the total cost came to exactly ₹750.
The rectangle below is as wide as the number of toys and as tall as the cost of each toy — so its area is the total cost. Drag the number of toys until the total is ₹750.
Toys made that day:
(ii) Let the number of toys made that day be .
Then the cost of each toy that day is
So the total cost that day is
Therefore , i.e. , i.e. , which we can write as
So the number of toys made that day satisfies the quadratic equation . Two different stories, two different quadratics — and in both of them the slider found two answers. Hold on to that clue.
Quadratic in Disguise
Some equations only look quadratic, and some quadratics come in disguise. The only way to be sure is to expand everything, bring all the terms to one side, and see whether an term survives.
Example 2 : Check whether the following are quadratic equations:
(i) (ii)
(iii) (iv)
Solution :
(i) The left-hand side is
So the equation becomes , i.e.
This is of the form , so the given equation
(ii) The left-hand side is , and the right-hand side is
So . The terms cancel, leaving
This is not of the form with , so the given equation
(iii) The left-hand side is
So , which gives
This is of the form , so the given equation
(iv) The left-hand side is
So . The terms cancel, leaving , or, dividing by 6,
This is of the form , so the given equation
Remark: don't judge an equation by its cover
Equation (ii) looks quadratic, but its terms cancel and it turns out to be linear. Equation (iv) looks cubic (degree 3), but its terms cancel and it turns out to be quadratic. So always simplify an equation first, and only then decide whether it is quadratic.
Speed Round: Quadratic or Not?
Expand in your head, cancel what cancels, then decide. After each answer you will see what the equation really simplifies to.
Get five right to move on.