Quadratic Equation

Equation ax2 + bx + c = 0
x ...
x ...
X2 + X + = 0

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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

How to use CalculatorHut’s Quadratic Equation Calculator?

Sometimes it is very difficult to calculate the roots since they may be in the imaginary form. So, to get the values of ‘x’ of any quadratic equation instantly, it is wise to go with this calculator. This saves lots of time and helps you to finish your job more efficiently. All you need to do is to enter the values of constants a, b, c and click Calculate to get the roots of the equation.

It’s so simple, isn’t it? Loved our quadratic equation calculator? Want to get the widget to your website and make calculations instantly?

Let us know your preferred customizations at [email protected], and we would design an attractive and colorful widget as per your taste. And all this is for FREE!!

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

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