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Quadratic Equations

    Quadratic Equations

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10th class > Quadratic Equations > Quadratic Equations

Quadratic Equations

A quadratic equation in the variable x is an equation of the form of ax2+bx+c=0, where a, b, c are real numbers, a  ≠ 0.

Example: 2x2+x−300=0 is a quadratic equation.

Similarly, 

2x2−3x+1=0,4x−3x2+2=0,1−x2+300=0 are also quadratic equations.

In fact, any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. But when we write the terms of p(x) in descending order of their degrees, then we get the standard form of the equation. That is, ax2+bx+c=0, a ≠ 0 is called the standard form of a quadratic equation.

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.

find out the marbles

  • Let the number of marbles John had be x .Then the number of marbles Jivanti had = – x
  • The number of marbles left with John, when he lost 5 marbles = x -
  • The number of marbles left with Jivanti, when she lost 5 marbles = 45-x- 5 then we get
  • Therefore, their product = (x – 5) (40 – x). Multiply both equations = x - x2 - + 5x
  • Therefore, the product is -x2+ x - 200
  • So, – x2+ 45x – 200 = 124 (Given that product = 124)
  • Therefore -x2 + 45x - =0
  • Therefore x2 - x + = 0
  • Therefore, the number of marbles John had, satisfies the quadratic equation x2 -45x +324 = 0 which is the required representation of the problem mathematically

(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.

find the number of toys

  • Let the number of toys produced on that day be x. Therefore, the cost of production (in rupees) of each toy that day = – x
  • So, the total cost of production (in rupees) that day = x (55 – x). Therefore, x (55 – x) =
  • mutltiply the eq with x ,therefore x - x2 =
  • Therefore −x2 + x - =0
  • Therefore x2 - x + = 0
  • Therefore, the number of toys produced that day satisfies the quadratic equation x2-55x+750 = 0 which is the required representation of the problem mathematically.

Check whether the following are quadratic equations:

(i)x−22+1=2x−3

check the eq is quadratic eq

  • LHS = x−22+ 1 = x2- + + 1
  • Now add the constant terms = x2 + 4x +
  • Therefore x−22+ 1 = 2x – 3 can be rewritten as
  • Therefore,add the values x2- x + = 0
  • It is of the form ax2 + bx +c = 0
  • Therefore, the given equation is a quadratic equation

Check whether the following are quadratic equations:

(ii) x(x + 1) + 8 = (x + 2) (x – 2)

check the eq is quadratic eq

  • Since x(x + 1) + 8 = x2 + + 8
  • and (x + 2)(x – 2) = x2 -
  • Therefore x2 + x + 8 = x2-
  • Therefore x + = 0
  • It is not of the form ax2 + bx + c = 0. Therefore, the given equation is not a quadratic equation.

Check whether the following are quadratic equations:

(iii) x (2x + 3) = x2 + 1

check the eq is quadratic eq

  • Here, LHS = x (2x + 3) = 2x2 + x
  • So, x (2x + 3) = x2 +
  • It can be rewritten as 2x2 + x = x2 +
  • Therefore, we get x2 + 3x - =
  • It is of the form ax2 + x + c =
  • So, the given equation is a quadratic equation.

Check whether the following are quadratic equations:

(iv) x+23 = x3 - 4

check the eq is quadratic eq

  • Here, LHS = x+23 = x3 + x2+x +
  • Therefore, x+23= x3-4 can be rewritten as
  • Therefore 6x2+ x + = 0
  • Or x2 + x + = 0
  • It is form of ax2+ bx + c = 0. So, the given equation is a quadratic equation.

Remark : Be careful! In (ii) above, the given equation appears to be a quadratic equation, but it is not a quadratic equation.

In (iii) above, the given equation appears to be a cubic equation (an equation of degree 3) and not a quadratic equation. But it turns out to be a quadratic equation. As you can see, often we need to simplify the given equation before deciding whether it is quadratic or not.