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 ax^{2}+bx+c = 0. Where a, b are the coefficients, c is constant and x is a variable. Here ‘a’ the coefficient of x^{2} cannot be equal to zero.

Example: 4x – 5 = 0. This can not be a quadratic equation since a = 0 and so the x^{2} term is absent. x^{2} – 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 b^{2} – 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: x^{2} – 5x + 4 = 0

x^{2 }– 4x – x + 4 = 0. Sum of (- 4x) + (- x) = -5x and the product of (- 4x) ( – x) = 4x^{2}

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: x^{2} – 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 ax^{2}+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 = ax^{2}+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. For more details about quadratic equations log on to the Cuemath website.

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