**The three ways of solving the quadratic equation**

The **Quadratic equations** are very helpful in calculating the unknown variable. If we know tendency of a variable how does it vary, we can form a **quadratic equation** and **solve** it to get the unknown value. Quadratic equations are very useful in science, maths, economics, business, and accounts. Here **three ways of solutions of a quadratic equation** are beautifully explained by an expert of maths by step by step method so every student can understand it easily.

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Studying our posts regularly will make you a hundred percent confident, enthusiastic, and conquerer. In this post, Future Study Point has brought the three ways of solving the quadratic equation. The solutions of questions are explained by three methods by the expert in such a way that even junior class students could understand it easily, though you are fully authorized to communicate with us by writing your comments in this blog, through e-mail, Facebook and WhatsApp if you face any problem.

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Before we go through the post, first of all, understand what is the quadratic equation. the algebraic expression in the form of ax² + ax + c is known as a quadratic expression or any algebraic expression in which highest power of the variable is 2 then such an expression is known as quadratic expression as an example 3x², 9x² +2, the equation involving quadratic expression is known as the quadratic equation,as an example ax² + ax + c =0

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The solution of this equation is determined by the factorizing quadratic expression thereby writing each factor equivalent to zero and then evaluating the value of the variable in the equation. The quadratic equation can be solved by factorization with the split-up method, By the complete square method and by using the quadratic formula See the example.

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**Example- Solve the quadratic equation 15x² + x – 6 = 0 .**

Ans. 15x² + x – 6 = 0

We can solve this equation by factorizing the expression 15x² + x – 6

We can solve it by choosing one of the ways of factorization.

**1 . SPLIT UP METHOD- **

Determine two factors of the product 15 × 6 = 90 such that the product of factor is 90 and their addition or difference is 1.

15x² + x – 6 = 0

15x² + (10 – 9)x – 6 = 0

15x² + 10 x– 9x – 6 = 0

5x(3x + 2) – 3(3x + 2)= 0

(3x + 2)(5x – 3) = 0

3x + 2 = 0

3x = – 2

5x – 3 = 0

5x = 3

Therefore the solutions of the given equation are 3/5 and –2/3

## 2. Factorising by complete square-

The given expression can be transformed into the complete square as follows

15x² + x – 6 = 0

Complete square mean writing the given expression, 15x² + x – 6 into the form of (a + b)²

Step1- Make the square term 15x² a complete square by multiplying 15 to the whole of the equation

we get , 225x² +15 x – 90 =0

Step2-Rewrite the equation as follows

(15x)² + 15x – 90 = 0

Step3- Comparing it to complete square (a² +2ab + b²)

If 15x→a, ⇒ 2ab = 15x ⇒ b = 15x/2a = 15x/30x =1/2

Step3-Rewrite the equation as follows by adding and subtracting (1/2)²

(15x)² + 15x + (1/2)² – (1/2)²– 90 = 0

(15x + 1/2)² – 1/4 – 90 = 0

(15x + 1/2)² – 361/4 = 0

(15x + 1/2)² –(19/2)² = 0

(15x + 1/2 + 19/2)(15x + 1/2 – 19/2) = 0

(15x + 10)(15x – 9) = 0

(15x + 10) = 0 ⇒ x = –2/3 and (15x – 9)=0⇒ x = 3/5

Therefore the solutions of the given equation are –2/3 and 3/5.

**3. Solution by the quadratic formula- **

The solution of the equation of the form ax² + bx + c , can be completed by the following formula.

We have the following equation

15x² + x – 6 = 0, where a = 15, b = 1 and c = –6

Placing values a,b and c in the above formula

Therefore the solutions of the given equations are –2/3 and 3/5.

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