Quadratic Equations

Synopsis

• An equation of the form ax²+bx +c = 0 is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.

Roots of the quadratic equation 

• The value of x that satisfies an equation is called the roots of the equation. 
• A real number a is said to be a solution/root of the quadratic equation ax² + bx + c = 0 if aa² + ba + c = 0.
• A quadratic equation has at most two roots

Solving Quadratic Equation by:

1. Splitting the middle term (or factorization) method
   If ax² + bx + c = 0, a ≠ 0, can be reduced to the product of two linear factors, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating each factor to zero.
   Steps involved in solving quadratic equationax² +bx+c = 0(a ≠0) by splitting the middle term (or factorization) method:
   Step 1: Find the product ac.
   Step 2: Find the factors of ‘ac’ that add up to b, using the following criteria:
   i.If ac > 0 and b > 0, then both the factors are positive.
   ii.If ac > 0 and b < 0, then both the factors are negative.
   iii.If ac < 0 and b > 0, then larger factor is positive and smaller factor is negative.
   iv.If ac < 0 and b < 0, then larger factor is negative and smaller factor is positive.
   Step 3: Split the middle term into two parts using the factors obtained in the above step.
   Step 4: Factorize the quadratic equation obtained in the above step by grouping method. Two factors will be obtained.
   Step 5: Equate each of the linear factors to zero to get the value of x.
   
   
2. Completing the square method
   Any quadratic equation can be converted to the form (x + a)² – b² = 0 or (x – a)² + b² = 0 by adding and subtracting the constant term. This method of finding the roots of quadratic equation is called the method of completing the square.
   The steps involved in solving a quadratic equation by completing the square, are as follows:
   Step 1: Make the coefficient of x² unity.
   Step 2: Express the coefficient of x in the form 2⨯ x ⨯  p.
   Step 3: Add and subtract the square of p.
   Step 4: Use the square identity (a + b)² or (a – b)² to obtain the quadratic equation in the required form (x + a)² - b² = 0 or (x – a)² + b² = 0.
   Step 5: Take the constant term to the other side of the equation.
   Step 6: Take the square root on both the sides of the obtained equation to get the roots of the given quadratic
   equation.
   
   
3. Quadratic formula
   The roots of a quadratic equation  ax² + bx + c = 0 (a ≠ 0) can be calculated by using the quadratic
   formula:
   
   If b² – 4ac < 0, then equation does not have real roots.

• Discriminant of a quadratic equation
  For the quadratic equation ax² + bx + c = 0, a ¹ 0, the expression b² – 4ac is known as discriminant.

• Nature of Roots
  Let f(x) = ax² + bx + c = 0 be the quadratic equation, the discriminant D = b² – 4ac.
  
  
  
  
• Relation between roots and coefficients:
  Let a, b  be the roots of the quadratic equation ax² + bx + c = 0  then

Note: Quadratic equation can be rewritten as.x2-(α+β) x+ α.β =0

• Equations reducible to Quadratic Equations
  There are many equations which are not in the quadratic form but can be reduced to the quadratic form by simplifications.
  Type I: ax²n + bxn + c = 0
  Put xn = y
  So, the equation reduces to  ay² + by +c = 0
  Now solve for y and hence for x.
  Type II:   where a, b, c are constants
  Make it  az² – cz + b = 0
  Now, solve it for z.
  
  Type III: (x+a) (x=b) (x+c) (x+d)+ k = 0,where the sum of two of the quantities   is equal to the sum of the other tw)o.
  Example:
  (x+1)(x+2)(x+3)(x+4) + 1= 0
  As sum of 1 and 4 is equal to sum of 2 and 3
  ⟹ [(x+1)(x+4)][(x+2)(x+3)+1 =0
  ⟹ (x² + 5x +4 )(x²+5x+6)+1=0
  Take  x²+5x =y
  ⟹ (y+4)(y+6)+1=0
  ⟹ y²+10y+24+1=0
  ⟹ y²+10y +25 =0
  ⟹ (y+5)² =0
  ⟹ y=-5
  ⟹ x² +5x+5=0
  
  

• Formation of Equations with given roots
  Suppose the given roots are α and β
  Therefore x = α and x =β
  ⟹ (x- α) = 0 and (x- β ) =0
  The equation forms will be
  (x- α)(x- β)=0
  ⟹ x² –( α+ β)x+ α β=0
  ⟹ x²-(sum of roots)x + Product of roots =0

• Applications of quadratic equations
• The applications of quadratic equation can be utilized in solving real life problems.
• Following points can be helpful in solving word problems:
• Every two digit number ‘xy’ where x is a ten’s place and y is a unit’s place can be expressed as. Xy= 10x +y.
• Downstream: It means that the boat is running in the direction of the stream
  Upstream: It means that the boat is running in the opposite direction of the stream
  Thus, if
  Speed of boat in still water is x km/h
  And the speed of stream is y km/h
  Then the speed of boat downstream will be (x + y) km/h and in upstream it will be (x − y) km/h.
  If a person takes x days to finish a work, then his one day's work