## Quadratic Equations - Summary

**①**A polynomial of degree 2 is called quadrilateral polynomial. The general form of a quadrilateral polynomial is ax

^{2}+ bx+ c, where a, b, c are real numbers such that a ≠ 0 and x is a real variable.

**②**If p(x) = ax

^{2}+ bx + c, a ≠ 0 is a quadratic polynomial and α is a real number, then p(α) = aα

^{2}+ bα + c is known as the value of the quadratic polynomial p(x).

③ A real number α is said to be a zero of the quadratic polynomial p(x) = ax

^{2}+ bx + c, if p(α) = 0.

④ If p(x) = ax

^{2}+ bx + c is a quadratic polynomial,then p(x) = 0 i.e. ax

^{2}+ bx + c = 0, a ≠ 0 is called a quadratic equation.

⑤ A real number α is said to be root of the quadratic equation ax

^{2}+ bx + c = 0, if aα

^{2}+ bα + c = 0.

In other words, α is a root of ax

^{2}+ bx + c = 0 if and only if α is zero pf the polynomial p(x) = ax

^{2}+ bx + c.

⑥ If ax

^{2}+ bc + c, a ≠ 0 is factorizable into product of two linear factors, then roots of the quadratic equation can be found by equating each factor to zero.

⑦ The roots of a quadratic equation can be found by using the method of completing the square.

⑧ The roots of the quadratic equation ax

^{2}+ bc + c, a ≠ 0 can be found by using the quadratic formula provided, b

^{2}- 4ac ≥ 0.

⑨ Nature of the roots of quadratic equation ax

^{2}+ bx + c = 0, a ≠ 0 depends upon the value of D = b

^{2}- 4ac, which is known as discriminate of the quadratic equation.

⑩ The quadratic equation ax

^{2}+ bx + c = 0, a ≠ 0 has:

⒜ Two distinct real roots, if D = b

^{2}- 4ac > 0

⒝ Two equal roots i.e. coincident real roots if D = b

^{2}- 4ac = 0

⒞ No real roots, if D = b

^{2}- 4ac < 0.

⑪ Greek mathematician Euclid developed a geometrical approach for finding out lengths

which in our present day terminology, are solutions of quadratic equations.

⑫ Solving quadratic equations in general form is often credited to ancient Indian Mathematicians. In fact,

**Brahma Gupta**(A.D 598 - 665) gave an explicit formula to solve a quadratic equation of the form ax

^{2}+ bx = c . Later

**Sridhar Acharya**(1025 A.D) derived a formula, now known as the quadratic formula, (as quoted by Bhaskara II) for solving a quadratic equation by the method of completing the square.

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