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Quadratic Equations

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

Quadratic Equation
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:
To solve a quadratic equation $ax^{2}+bx+c=0$ by this method, we follow the following steps:
  • 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.
For Example: $3x^{2}+2x-1=0$
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:
Quadratic equations can be solved by completing the perfect square and using the suitable identities.
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 $a^{2}-b^{2}=(a+b)(a-b)$)
$x = -2+\sqrt{3}$
$x = -2-\sqrt{3}$

(3) Quadratic Formula: We can use the following quadratic formula to find the solution of a quadratic equation -
Quadratic Formula
For Example: $x^{2}+x-1=0$
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}$

Related Calculators
C Quadratic Equation Solver Factor Quadratic Equation Calculator
Graphing Quadratic Equations Calculator
 

More topics in  Quadratic Equations
Factoring Quadratic Equations Graphing Quadratic Functions
Graphing Quadratic Equations How to Solve Quadratic Equations by Graphing
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