QUADRATIC EQUATIONS Explained

An equation which involves the term x² is known as a quadratic equation.

A quadratic equation:

• Should have a term with the maximum power 2, e.g., 3x², −5x², x²
• May also contain terms involving x, such as 5x, −7x, or 0.5x.
• May also have constant terms which are just numbers like 4, −8, 112.

Note:

A quadratic equation CANNOT have terms involving higher powers of x, like x³ and also it cannot have terms like 1/x in it.

General representation of a quadratic equation

The general representation of a quadratic equation is represented as ax² + bx + c = 0 where ‘a’ can be any number excluding zero. The value of ‘b’ and ‘c’ can be any numbers including zero. If b or c is zero, then these terms will get eliminated.

An equation with variable x of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0 is known as quadratic equation.

Roots of a quadratic equation

A number α is said to be a root of the quadratic equation ax² + bx + c = 0, if aα² + bα + c = 0.

The roots of the quadratic equation ax² + bx + c = 0 are the same as the zeroes of the quadratic polynomial equation ax² + bx + c.

Methods of Finding Roots of Equations

We can find the value of the variable ‘x’ by using various methods.

• Factorization method
• Completing the square
• Using formula
• Using graphs

Among all the above methods, Factorization and formula method are the most frequently used.

FACTORIZATION METHOD

If we can factorize the quadratic polynomial ax² + bx + c, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating the linear factors of ax² + bx + c to zero.

Example

Assume a quadratic equation x² − 5x + 6 = 0.

We factorize the quadratic by looking for two numbers which multiply together to give 6, and add to give −5.

Now −3 × −2 = 6 and− 3 + −2 = −5. So, the two numbers are −3 and −2. We use these two numbers to write −5x as −3x − 2x and proceed to factorise as follows:
x² − 5x + 6 = 0
x² − 3x − 2x + 6 = 0
x(x − 3) − 2(x − 3) = 0
(x − 3)(x − 2) = 0
(x – 3) = 0 or (x – 2) = 0
So we get x = 3 or x = 2.

QUADRATIC FORMULA

If b² – 4ac ≥ 0, then the real roots of the quadratic equation ax² + bx + c = 0 are given by (−b ± √ (b² − 4ac))/2a.

The expression ‘b²– 4ac calculates the discriminant of the quadratic equation’. Discriminant plays an important role in knowing the nature of the roots. The quadratic equation (ax²+bx+c=0) roots can be classified as following:

1. roots are real and distinct if b² – 4ac > 0
2. roots are real and equal if b² – 4ac = 0
3. roots are imaginary if b² – 4ac < 0

Assume the quadratic equation x² − 3x − 2 = 0.

Comparing the above equation with the general form ax² + bx + c = 0

We observe that a = 1, b = −3 and c = −2.

These values are substituted into the formula.
x = (−b ± √ (b² − 4ac))/2a
= (3 ± √ (9 + 8)) /2
= (3 ± √ 17)/2

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Life is a Math Equation. To gain the most, you have to Know How to convert Negatives into Positives.