Quadratic equation is the polynomial equation of degree "two" in one variable. More precisely, we can say that an equation of the form

is known as quadratic equation.

a, b and c are constants and x is the variable.

"a" must not be zero because if "a" equals zero, the equation will be no more an equation of degree two.

Quadratic equations can be solved by various methods. By solving a quadratic equation, we mean finding the value of variable contained in it.

Methods for Solving Quadratic Equations

(1) Factorization:

- Determine the value of a, b and c.
- Calculate the product a*c.
- Split the value of "b" in two parts so that their sum or difference is equal to "b" and product is equal to "ac".
- Substitute the split parts in place of b.
- Separate like terms and take common factors out.

a = 3, b = 2, c = -1

a * c = -3 = 3 * (-1)

b = 2 = 3 - 1

$3x^{2}+(3 - 1)x-1=0$

$3x^{2}+3x-x-1=0$

3x (x + 1) - 1(x + 1) = 0

(x + 1) (3x - 1) = 0

x = -1, $\frac{1}{3}$

(2) Completing the Square:

For example: $x^{2}+4x+1=0$

$x^{2}+4x+4-4+1=0$

$(x^{2}+4x+4)-4+1=0$ (we know that $(a+b)^{2}=a^{2}+2ab+b^{2}$)

$(x+2)^{2}-3=0$

$(x+2)^{2}-(\sqrt{3})^{2}=0$

$(x+2+\sqrt{3})(x+2-\sqrt{3})=0$ (we have $a^{2}-b^{2}=(a+b)(a-b)$)

$x = -2+\sqrt{3}$

$x = -2-\sqrt{3}$

Here, a = 1, b = 1, c = -1

$x=$$\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$

$x=$$\frac{-1\pm \sqrt{1+4}}{2}$

$x=$$\frac{-1\pm \sqrt{5}}{2}$

=> $x=$$\frac{-1+ \sqrt{5}}{2}$ and $x=$$\frac{-1- \sqrt{5}}{2}$

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