### Quadratic Equations

In this tutorial, you will learn:

- What a quadratic equation is
- How to solve by factorisation
- How to solve by quadratic formula
- How to solve quadratic equations by completing the square
- How to solve a quadratic equation graphically
- How to generate quadratic equations randomly by using a programme, at the end of the tutorial for practice
- How to use Microsoft Excel in solving quadratic equations by using the formula method

#### Quadratic Equations

An equation in the form of ax^{2} + bx + c = 0 is called a quadratic equation.

- x
^{2}+ 6x + 8 = 0 - 2x
^{2}- 5x + 6 = 0 - x
^{2}- 6 = 0 - x
^{2}- 6x = 0

A quadratic equation has two solutions; that means there are two values for x that satisfy the equation. There are four different ways to solve a quadratic equation:

- Factorizing Method
- Formula Method
- Graphical Method
- Completing the Square Method

#### Factorizing

**E.g.1**

x^{2} + 8x = 0

x(x + 8) = 0

x = 0 or (x + 8) = 0

x = 0 or x = -8

**E.g.2**

x^{2}= 6x

x^{2} - 6x = 0

x(x - 6) = 0

x = 0 or (x - 6) = 0

x = 0 or x = 6

**E.g.3**

x^{2} + 6x + 8 = 0

x^{2} + 4x + 2x + 8 = 0

x(x + 4) + 2(x + 4) = 0

(x + 4)(x + 2) = 0

(x + 4) = 0 or (x + 2) = 0

x = -4 or x = -2

**E.g.4**

x^{2} - 6x + 8 = 0

x^{2} - 4x - 2x + 8 = 0

x(x - 4) - 2(x - 4) = 0

(x - 4) = 0 or (x - 2) = 0

x = 4 or x = 2<

**E.g.5**

^{2}+ 6x - 16 = 0

x

^{2}+ 8x - 2x - 16 = 0

x(x + 8) - 2(x + 8) = 0

(x + 8) = 0 or (x - 2) = 0

x = -8 or x = 2

**E.g.6**

2x^{2} + 13x + 6 = 0

2x^{2} + 12x + x + 6 = 0

2x(x +
6) + 1(x + 6) = 0

(x + 6) = 0 or (2x + 1) = 0

x = -6 or 2x = -1

x = -6 or x = -1/2

**E.g.7**

x^{2} - 9/4 = 0

(x + 3/2)(x - 3/2) = 0

x + 3/2 = 0 or x -
3/2 = 0

x = -3/2 or x = 3/2

#### Formula Method

If ax^{2} + bx + c = 0, then

**x = [-b ±√(b ^{2} -
4ac) ]/ 2a**

**E.g.1**

x^{2} - 6x + 8 = 0

a = 1; b = -6; c = 8

x = -(-6) ±√((-6)^{2}
- 4(1)(8)) / 2(1)

x = 6 ±√(36 - 32) / 2

x = 6 ±√(4) / 2

x = (6
± 2 )/ 2

x = 4 or x = 2

**E.g.1**

2x^{2} - 5x + 3 = 0

a = 2; b = -5; c = 3

x = -(-5) ±√((-5)^{2}
- 4(2)(3)) / 2(2)

x = 5 ±√(25 - 24) / 4

x = 5 ±√(1) / 4

x = (6
± 1 )/ 4

x = 1.5 or x = 1

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#### Graphical Method

In this method, a graph is plotted for a quadratic function. The graph takes the
typical shape, known as ** parabola**.

E.g. Solve x^{2} + 5x - 7 = 0

First of all, make a table for both x and y of the function.

x | y |

-2 | -13 |

-1 | -11 |

0 | -7 |

1 | -1 |

2 | 7 |

3 | 17 |

Now, plot a graph of y against x. Note the points at which the curve the crosses
the x-axis. They are the solutions of the quadratic function. The solutions are:

x = 1.1 or x = -6.1

The following animation is interactive: it shows how to solve a quadratic equation by a graph; by clicking on the button, you can generate a random equation and its solutions appear at the same time. If there are no solutions - the graph being above the x-axis - instead of solutions, the word, *undefined*, appears in those places.

#### Completing the Square Method

x^{2} + 4x - 5 = 0

Let (x + a)^{2} + b = x^{2} +
4x - 5

x^{2} + 2ax + a^{2} + b = x^{2} + 4x - 5

Now, make the coefficients of x and the constant equal.

x => 2a = 4

a =2

a^{2} + b = -5

4 + b = -5

b = -9

(x + 2)^{2}
- 9 = 0

(x +2)^{2} = 9

(x + 2) = ±3

x = -2 ±3

x
= 1 or -5

#### Solving Quadratic Equations with MS Excel

The following animation shows how MS Excel can be easily programmed to solve quadratic equations. It is done by **VBA** code, which is fairly simple to grasp.

**Now, in order to complement what you have just learnt, work out the
following questions:**

Click the button to get the quadratic equations; solve them by all four methods to master the techniques.

**Quadratic Equation Generator**