This is one of the mathematical equation-based questions where variables and numbers are given. **4x ^ 2 – 5x – 12 = 0** is an example of a quadratic equation. This equation is a mathematical equation and it requires a formula to solve it.

This equation will test your mathematical knowledge and abilities. Such forms of mathematics improve ours thinking ability. This type of quadratic equations problem is generally solved by simplification method. Simplification method means breaking of equations till it equalizes the equation. In this equation.

**About Quadratic Equation: 4x ^ 2 – 5x – 12 = 0**

It is a problem-based equation that is asked to be solve to prove a solution. It is derived from a Latin word quadratus. We have to prove that left hand side is equal to right hand side** (L.H.S = R.H.S). **Mathematical equations are just seemed to be complex but they’re actually difficult.

Quadratic equation is a mathematical term. The given equation is a prime example of quadratic equation. Quadratic equations are represented in the form of ax^2 + bx + c = 0.

Example: In the given equation, 4x ^ 2 – 5x – 12 = 0

The term **a, b, c** are known as coefficient, a is not equal 0. The term ‘**x’** is represented as variable.

**Two Forms of Quadratic Equation**

**Criterion**

This form is represented by **y = ax ^{2} + bx + c,** where a, b, c are numbers and x is a variable.

**Factored Form**:

This form is represented by **y = ( ax + c ) ( bx + d** , where a, b, c, and d are just numbers.

**Explain**:- **Why coefficient ‘a’ is not equal to zero in a quadratic equation?**

If the coefficient ** ‘a’** is represented with value 0 then it will change the whole form of equation. It will get converted to a linear equatio**n** from the quadratic equation. ** **

It is a theory that coefficient ‘a’ can never be represented with a value 0 or else will never find a solution. Mathematics have rules and regulations for each mathematical concept.

**Formula to solve = 4x ^ 2 – 5x – 12 = 0**

The given equation is a quadratic expression. It requires a specific formulas to solve such the world of mathematical expressions.

**Quadratic Formula: x = [(-b ± √(b ^{2} – 4ac)) / 2a]**

Through above formula we can easily calculate the method to solve the given equation, 4x ^ 2 – 5x – 12 = 0.

This formula is a formula of simplification in which we simplify the equation as much as possible. Al-Khwarizmi gave this formula to solve all the equations to its simplest form.

**Techniques to** ** Solve ****4x ^ 2 – 5x – 12 = 0**

**4x ^ 2 – 5x – 12 = 0**

**Completing the square**

Divide the equation by the coefficient of square variable. It means divide the equation by the coefficient ‘a’ then imply the square method. Add a constant term on both the sides of the equation and calculate.

**Factoring**

Factoring is a technique to solve an equation by splitting the middle terms of the equation. Factoring technique is used to determine the factors of an equation and then it is solved.

**Taking the Square Root**

In this technique we have separated the square terms and the constant term opposite to each other. Square root out is a method to find a solution for solving Quadratic Equation. In this technique we solve the roots of both the sides to get an accurate answer for this.

**Using Quadratic Formula**

x = [(-b ± √(b^{2} – 4ac)) / 2a], this is a quadratic formula, by simply adding all the respective values in this formula we can find a right solution.

**Answer – Solution of the 4x^2 – 5x – 12 = 0**

We will solve the equation by quadratic formula.

Quadratic formula: x = [(-b ± √(b^{2} – 4ac)) / 2a]

In this given equation**: a = 4, b = – 5 and c = – 12 **where a is not equal to 0. The term ‘x’ is an unknown factor.

To solve the given problem we need to place the values on the quadratic formula by Al-Khwarizmi.

**Assume, X = x**

**X = [ -b ± √ (b**^{2}– 4ac)] / 2a**X = [ – (-5) ± √ ( (-5)**^{2}– 4 (4) (-12))] / 2 (4)**X = [ 5 ± √ ( 25 +192 )] / 8****X = 5 + √ 217 / 8 and X = 5 – √ 217 / 8**

**Thus, we got the solution where, x = 2.466 and x = – 1.216**

Two consecutive solution for the equation are: x = 2.466 and x = – 1.216

**Similar equation to 4x ^ 2 – 5x – 12 = 0**

*List of examples of quadratic equation. These are three examples of what quadratic equations looks like. You can give it a try to solve by applying the above given formula.*

- 6×2 + 11x – 35 = 0
- 2x
^{2}– 4x -2 = 0 - 2x
^{2}+ 4x – 5 = 0

These equations are further solved with the help of quadratic formula to find the value of ‘x’.

**Graph x = 2.466 and x = – 1.216**

**Key Points**

- Quadratic formula:
**x = [ -b ± √ (b2 – 4ac)] / 2a**. In this given equation: a = 4, b = – 5 and c = – 12 where a is never or not equal to 0. - Quadratic equation is a mathematical term. It is a problem-based equation that is asked to be solve to prove a solution.
- The equation has been represented into tow forms. The given problem was in the Standard quadratic form which was simplified to the Factored form to calculate the solution.
- Taking the square root , Factoring , Quadratic formula and Completing the formula are the methods to solve quadratic problems.
- The two consecutive solution for the equation 4x^2 – 5x – 12 = 0 are:
**x = 2.466 and x = – 1.216**Generally, in the mathematical world we avoid negative value of the x to get the right answer.

## Conclusion

Mathematics is a game of formula, if you know the formula you can crack all the mathematical problems. Quadratic equations are very easy to solve when we apply the right formula**.** If we have the formula, we have just put the right value at the right position and it’s done.

This equation can be solve by four different methods. In the equation can be asked in the format of multiple choice questions also. By breaking down the value of ‘x’ we have found the right solution for the term.

To find the solution for the above listed three equations, same quadratic formula is used to find the value of ‘x’. Mathematics just looks difficult but it’s not a difficult subject. Quadratic equations is a small universe in the cosmos of mathematics.

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