Equation | ax^{2} + bx + c = 0 |

x | ... |

x | ... |

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= 16

= 16

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#
QUADRATIC EQUATIONS Explained

#### General representation of a quadratic equation

#### Roots of a quadratic equation

#### Methods of Finding Roots of Equations

#### FACTORIZATION METHOD

#### QUADRATIC FORMULA

#### How to use CalculatorHut’s Quadratic Equation Calculator?

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An equation which involves the term x^{2} is known as a quadratic equation.

A quadratic equation:

- Should have a term with the maximum power 2, e.g., 3x
^{2}, −5x^{2}, x^{2} - 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 x^{3} and also it cannot have terms like 1/x in it.

The general representation of a quadratic equation is represented as ax^{2} + 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 ax^{2} + bx + c = 0, where a, b, c are real numbers and a ≠ 0 is known as quadratic equation.

A number α is said to be a root of the quadratic equation ax^{2} + bx + c = 0, if aα^{2} + bα + c = 0.

The roots of the quadratic equation ax^{2} + bx + c = 0 are the same as the zeroes of the quadratic polynomial equation ax^{2} + bx + c.

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.

If we can factorize the quadratic polynomial ax^{2} + bx + c, then the roots of the quadratic equation ax^{2} + bx + c = 0 can be found by equating the linear factors of ax^{2} + bx + c to zero.

Example

Assume a quadratic equation x^{2} − 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:

x^{2} − 5x + 6 = 0

x^{2} − 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.

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

The expression ‘b^{2}– 4ac calculates the discriminant of the quadratic equation’. Discriminant plays an important role in knowing the nature of the roots. The quadratic equation (ax^{2}+bx+c=0) roots can be classified as following:

- roots are real and distinct if b
^{2}– 4ac > 0 - roots are real and equal if b
^{2}– 4ac = 0 - roots are imaginary if b
^{2}– 4ac < 0

Assume the quadratic equation x^{2} − 3x − 2 = 0.

Comparing the above equation with the general form ax^{2} + bx + c = 0

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

These values are substituted into the formula.

x = (−b ± √ (b^{2} − 4ac))/2a

= (3 ± √ (9 + 8)) /2

= (3 ± √ 17)/2

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.

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