Solving Quadratic Equations

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The polynomial equation of second degree is called a quadratic equation.

The general form of quadratic equation is ax2 + bx + c = 0, where a, b, c are real numbers and.

We can solve quadratic equations by following methods –

1- Factorization

2- Completing the square

3- Using the quadratic formula

1) Factorization

In factorization we split the middle term into two parts such that the product of first term and the last term is equal to the product of the two splitted parts, then we perform grouping and at last we put both the brackets equal to zero to find out the required solutions.

Exampley2 + y – 2 = 0

                         Finding solutions of the given equation

                         y2 + 2yy – 2 = 0 (Splitting the middle term)

                         y(y + 2) – 1(y + 2) = 0

                         (y + 2) (y – 1) = 0 (Grouping)

                          Either (y + 2) = 0 or (y – 1) = 0

                           y = -2 or y = 1 are the required solutions.

2) Completing the square –

First we make coefficient of x2 as 1 by dividing the equation by a, then bring constant to the right side and then making left side perfect square by adding suitable constant both sides. After that we will take square root of both sides and finally we will solve for x.

  Example s2-3s-10=0

                      s2-3s=10

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Using the Quadratic Formula:

First, compare the given equation with the general equation ax2 + bx + c, then put the values of a, b, c in the quadratic formula

Example -2x2+5x-6=0

 Putting value after comparing with general equation

Some facts about Quadratic Equations:

1) Any number when putting in place of variable if satisfies the equation is called the root of the equation.

2)  Zeros of  ax2 + bx + c and roots of ax2 + bx + c=0  are same.

 

Check point

Solve the following quadratic equations-

1) x2 + 15x + 56 = 0

2) a2 + 9a + 20 = 0

3) b2 + 2b – 4 = 0

4) y-(6/y) =5

5) p2 – 5p + 6 = 0

 

Answer Key

1)   -7, -8

2)    -5, -4

3)    -1+ squareroot(5), -1-squareroot(5)

4)      6, -1

5)       2, 3

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