According to Wikipedia, “a quadratic equation is a equation that can be arranged in standard form as

ax^2 + bx + c = 0

Where x is unknown and a, b and c are known coefficients. The coefficient terms a, b and c are also known as quadratic coefficient, linear coefficient and constant terms respectively. The condition to be quadratic is a\neq 0 . If a=0, the equation becomes linear”. In solving quadratic equation we need to determine the value of unknown (x in this case). Now we will see how to find value of x

Factorization Method:

Let’s take a quadratic equation with known coefficients as

x^2 + 2x - 24 = 0

Steps:

• Multiply the constant coefficients a and c with their sign. Here  a = 1 and c = -24 , so multiplication results = -24
• Factorize the magnitude of multiplication result, which is 24. Factorization need not necessarily be prime factor. 24 = 2\times 2\times 2\times 3
• Take any combinations of these factors to make two such components that their addition (with sign) equals to bx (here 2x)  and their multiplication equals to ac (here -24x²). So, we can write 2x = 6x - 4x   (multiplication of 6x and -4x = -24x²)

Now we can rewrite the equation as

x^2 + 6x - 4x - 24 = 0

Now take two of the terms with the common term being separated as multiplying factor , for example, in x²+6x, x is common in both terms so that we can write it x(x+6). The above equation now becomes,

x(x+6) -4(x+6) = 0

Again separate the common factor in the above equation which gives,

(x+6)(x-4) = 0

The product of two numbers is equal to zero when any of the term is zero. So, we can write

x+6 = 0 \: \text{or}, x = -6

and x-4 = 0 \: \text{or}, x = 4

Putting the value of x equal to -6 and 4 in the equation, we find that both value satisfy the equation. So the solution is x = -6, 4

Square Root Method:

Let the equation be x^2 - 6x - 7 = 0

The above equation can be written as,

x^2 - 6x + 9 - 16 = 0 \\

\text{or,} (x-3)^2 = 16 \\

\text{or,} x-3 = \pm \sqrt{16} \\ \\

So, x-3 = 4 \: \text{or,} x = 7 \\

and x-3 = -4 \: \text{or,} x= -1

Quadratic Formula Method:

The formula for solving quadratic equation ax^2 + bx + c = 0 is given by

\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

So, quadratic equation has two roots as,

x = -\frac{b}{2a} + \frac{\sqrt{b^2-4ac}}{2a} \: \: \text{and} x = -\frac{b}{2a} - \frac{\sqrt{b^2-4ac}}{2a}

The roots have the following properties:

• If b^2 - 4ac > 0 roots are real and distinct
• If b^2 - 4ac = 0 roots are identical
• If b^2 - 4ac < 0 roots are complex

Example:

Solve the equation: 3x^2 + 5x - 1 = 0

Here, a = 3, b= 5 and c= -1

So, x = \frac{-5 \pm \sqrt{25+12}}{6} \\

or, x = \frac{-5 + \sqrt{37}}{6} \: \: \text{and} \frac{-5 - \sqrt{37}}{6}

Practical Example:

Suppose that you have a rectangular land lot which length is 30 feet longer than width. The area of the plot is 1000 square feet. What is the length and width of the plot?

Let, the width be x

So, the length = x+30

Now, x(x+30) = 1000 \\

or, x^2 + 30x - 1000 = 0 \\

or, x^2 + 50x - 20x - 1000 = 0 \\

or, x(x+50) - 20(x+50) = 0 \\

or, (x+50)(x-20) = 0 \\

So, x = -50, 20

Since width cannot be negative, the solution is width = 20 feet and length = 20+30 = 50 feet