Solving a Quadratic equation

You might have heard of different types of equations like linear, quadratic, cubic, etc. All these equations might have messed up in your head. Not to worry, solving quadratic equations can help you in clearing the confusion. In algebra, a Quadratic equation is a second-degree equation, since the exponent of the variable is 2. It has got numerous applications in physics, engineering, etc. 

Standard Form of Quadratic Equations:

Any equation that can be rearranged in the following standard form is known as a quadratic equation.

The standard form is ax2+bx+c = 0. Where a, b are the coefficients, c is constant and x is a variable. Here ‘a’ the coefficient of x2 cannot be equal to zero. 

Example: 4x – 5 = 0. This  can not be a quadratic equation since a = 0 and so the x2 term is absent. x2 – 4x + 5 = 0. This is a quadratic equation with a value of a = 1.

Formula to Solve a Quadratic Equation:

There is a standard formula to solve a quadratic equation that is, 

x = -b b2 – 4ac         2a

The formula is used to find the roots or zeros of a quadratic equation. Roots are nothing but solutions found for the quadratic equations. We get two solutions for a quadratic equation.

The value obtained by solving only b2 – 4ac is called the discriminant of a quadratic equation. Based on this value we predict the nature of the roots. That is if the value of discriminant > 0 then the roots are real and distinct. If the value of discriminant = 0 then the roots are real and equal.If the value of discriminant< 0 then the roots do not exist or the roots are imaginary.

Methods to Solve a Quadratic Equation:

A system of equations is a set of equations that can be solved together. One can solve both linear and quadratic equations together. In that case, it is called a system of linear and quadratic equations.

A quadratic equation can be solved using any of the following methods.

1. Factorization: It’s a technique used to find the roots of a quadratic equation. Here the middle term of the equation is split into 2 terms, such that the product of these terms gives you the product of the first and the last term.

Example: x2 – 5x + 4 = 0

x2 – 4x – x + 4 = 0. Sum of (- 4x) + (- x) = -5x and the product of (- 4x) ( – x) = 4x2

Now taking x – 4 as common 

x(x – 4) – 1(x – 4) = 0

(x – 4) (x – 1) = 0

∴ x = 4 and x = 1. 

2. Using the standard formula: Let us solve the same equation using the above mentioned standard formula

Example: x2 – 5x + 4 = 0

The standard formula to solve a quadratic equation is, 

x = -b b2 – 4ac         2a

Here a = 1, b = -5 and c = 4

By substituting these values in the above equation we get,

x = -( – 5) (-5)2 – 4(1)(4)         2(1)

x = 5 25 – 162 = 5 32 

x = 4 or x = 1

3. The method of completing the squares:

In this method, we reduce the standard equation to get the roots then by substituting the values we can get the final values of the roots.

That is ax2+bx+c = 0 is simplified to get x = -b b2 – 4ac         2a then by substituting the values we get the final roots.

4. Graphing method: In this method, the graph is plotted by considering the standard equation as a function of Y. i,e. Y = ax2+bx+c. For values of x and y, the graph is plotted. The points at which the graph cuts the x-axis give the roots of the equation.

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