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• Quadratic is derived from a latin word quadratus. This means square.  Quadratic definition refers to an expression that involves the variable of an order 2.
• Quadratic expression has variables that have the maximum possible power / exponent / or degree that is  two. Quadratic equations are the polynomial equations of second degree.

History of quadratic equations and methods to solve them:

• Scientist Pythagoras and Euclid together formulated techniques and methodology to solve quadratic equations.

• It is the homogeneous equation that has variable with the maximum degree two.

Generically a quadratic equation is written as

ax^2 + bx+ c=0

• When this equation is solved by using completing square technique we get a formula which is called quadratic formula
• This formula is used to get the easy solution of variable involved in a quadratic equation by simply placing the values of numeric coefficients involved named by a and b and also numeric constant c.
• The factorization method may not always be successful so this formula was generated to find solution of quadratic expressions. Every quadratic equation has only two possible solutions. These solutions are called roots.

Discriminant:

• In order to find the nature of roots of a quadratic equation we have to calculate the value of discriminant first
• Discriminant is (b2−4ac)
• If  (b2−4ac)=0

Roots of given equation are real and equal

• If (b2−4ac)>0

Roots are real unequal

• And now if discriminant is a perfect square roots are rational if it is not a perfect square roots are irrational
• If (b2−4ac)<0

Roots of the given quadratic equation are complex/imaginary and unequal as well.

Graph of a quadratic equation is drawn by the form of parabola. If the parabola opens upwards the vertex of parabola have minimum value
In the origin  x and y vertices are of value (0,0)

The vertex is the lowest point on the parabola if it opens upwards. While if it opens downwards vertex would be the highest point. The line drawn vertical is called as the line of symmetry that divides parabola into two equal halves for x=c

Graph of y = 2x^2 - 12x + 7

Completing square:
Starting with a quadratic equation in standard form, ax2 + bx + c = 0

1. Divide each side by a, the coefficient of the squared term.
2. Subtract the constant term c/a from both sides.
3. Add the square of one-half of b/a, the coefficient of x, to both sides. This "completes the square", converting the left side into a perfect square.
4. Write the left side as a square and simplify the right side if necessary.
5. Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
6. Solve the two linear equations.

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