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Quadratic Equations

Solving Quadratic Equations

We saw that quadratic equations can represent many real-life situations. Now that we know what quadratic equations are, let us learn about the different methods to solve them. Here we will try to develop the Quadratic Equation Formula and other methods of solving the quadratic equations. Let us start!

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Introduction to Quadratic Equations
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Methods of Solving Quadratic Equations

There are three main methods for solving quadratic equations:

  • Factorization
  • Completing the square method
  • Quadratic Equation Formula

In addition to the three methods discussed here, we also have a graphical method. As you may have guessed, it involves plotting the given equation for various values of x. The intersection of the curves thus obtained with the real axis will give us the solutions. Let’s see the others in detail.

Factorization

The first and simplest method of solving quadratic equations is the factorization method. Certain quadratic equations can be factorised. These factors, if done correctly will give two linear equations in x. Hence, from these equations, we get the value of x. Let’s see an example and we will get to know more about it.

Examples of Factorization

Example 1: Solve the equation: x2 + 3x – 4 = 0
Solution: This method is also known as splitting the middle term method. Here, a = 1, b = 3, c = -4. Let us multiply a and c = 1 * (-4) = -4. Next, the middle term is split into two terms. We do it such that the product of the new coefficients equals the product of a and c.

We have to get 3 here. Consider (+4) and (-1) as the factors, whose multiplication is -4 and sum is 3. Hence, we write x2 + 3x – 4 = 0 as x2 + 4x – x – 4 = 0. Thus, we can factorise the terms as: (x+4)(x-1) = 0. For any two quantities a and b, if a×b = 0, we must have either a = 0, b = 0 or a = b = 0.

Thus we have either (x+4) = 0 or (x-1) = 0 or both are = 0. This gives x+4 = 0 or x-1 = 0. Solving these equations for x gives: x=-4 or x=1. This method is convenient but is not applicable to every equation. In those cases, we can use the other methods as discussed below.


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Completing the Square Method

Each quadratic equation has a square term. If we could get two square terms on two sides of the quality sign, we will again get a linear equation. Let us see an example first.

Example 2: Let us consider the equation, 2x2=12x+54, the following table illustrates how to solve a quadratic equation, step by step by completing the square.
Solution: Let us write the equation 2x2=12x+54. In the standard form, we can write it as: 2x2 – 12x – 54 = 0. Next let us get all the terms with x2 or x in them to one side of the equation: 2x2 – 12 = 54

In the next step, we have to make sure that the coefficient of x2 is 1. So dividing throughout by the coefficient of x2, we have: 2x2/2 – 12x/2 = 54/2 or x2 – 6x = 27. Next, we make the left hand side a complete square by adding (6/2)2 = 9 i.e. (b/2)2 where ‘b’ is the new coefficient of ‘x’, to both sides as: x2 – 6x + 9 = 27 + 9 or x2 – 2×3×x + 32 = 36. Now we can write it as a binomial square:

  • (x-3)2 = 36;    Take square root of both sides
  • x – 3 = ±6;      Which gives us these equations:
  • x = (3+6)    or   x = (3-6) or x = 9 or x = -3

This is known as the method of completing the squares.

Quadratic Equation Formula

There are equations that can’t be reduced using the above two methods. For such equations, a more powerful method is required. A method that will work for every quadratic equation. This is the general quadratic equation formula. We define it as follows: If ax2 + bx + c = 0 is a quadratic equation, then the value of x is given by the following formula:

Quadratic Formula

Just plug in the values of a, b and c, and do the calculations. The quantity in the square root is called the discriminant or D. The below image illustrates the best use of a quadratic equation.

Example 3: Solve: x2 + 2x + 1 = 0

Solution: Given that a=1, b=2, c=1, and
Discriminant = b2 − 4ac = 22 − 4×1×1 = 0
Using the quadratic formula,  x = (−2 ± √0)/2 = −2/2
Therefore, x = − 1

More Solved Examples For You

Question 4: Find the value of x:  27x2 12 = 0
A) 2/3    B) ± 2/3     C) Ambiguous    D) None of these

Answer : B) Here, a =  27, b = 0 and c = -12. Hence, from the quadratic formula, we have:
x = − 0 ± √02 – 4(27)(-12)/2 (27)
Thus x = ± √(4/9) = ± 2/3

Question 5: What is the formula for solving quadratic equation?

Answer: The general quadratic equation formula is “ax2 + bx + c”. In this formula, a, b, and c are number; they are the numerical coefficient of the quadratic equation and ‘a’ is not zero a  0.

Question 6: What is quadratic equation?

Answer: Simply, a quadratic equation is an equation of degree 2, mean that the highest exponent of this function is 2. Moreover, the standard quadratic equation is ax2 + bx + c, where a, b, and c are just numbers and ‘a’ cannot be 0. An example of quadratic equation is 3x2 + 2x + 1.

Question 7: Name of different method by which you can solve quadratic equation?

Answer: There are various methods by which you can solve a quadratic equation such as: factorization, completing the square, quadratic formula, and graphing. These are the four general methods by which we can solve a quadratic equation.

Question 8: What are the three form of quadratic function?

Answer: The three function are listed below which can be written as:

  1. Standard Form: y = ax2 + bx + c, here a, b, and c are just numbers.
  2. Factored Form: y = (ax + c) (bx + d) here also a, b, and c are just numbers.

Vertex Form: y = a (x + b)2 + c, and here also a, b, and c are numbers.

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vee kwari

how do we state the nature of quadratics

Karthik
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Karthik

A polynomial can have real numbers as zero(ie rational and irrational)to decide it’s nature we can use the relationship between the coefficients and zero (by comparing sum and product of zeroes)

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Vikas

HI

HELLOOO
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Hi

Hjgv
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Hjgv

If d 0 – real & distinct roots
d =0 – real & equal roots ….

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