This is known as the method of completing the squares.
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(x-3) 2 = 36 Take square root of both sides.Now we can write it as a binomial square: (b/2) 2 where ‘b’ is the new coefficient of ‘x’, to both sides as: x 2 – 6x + 9 = 27 + 9 or x 2 – 2×3×x + 32 = 36. Next, we make the left hand side a complete square by adding (6/2) 2 = 9 i.e. So dividing throughout by the coefficient of x 2, we have: 2x 2/2 – 12x/2 = 54/2 or x 2 – 6x = 27. In the next step, we have to make sure that the coefficient of x 2 is 1. In the standard form, we can write it as: 2x 2 – 12x – 54 = 0. Next let us get all the terms with x 2 or x in them to one side of the equation: 2x 2 – 12 = 54 Solution: Let us write the equation 2x 2=12x+54. Let us see an example first.Įxample 2: Let us consider the equation, 2x 2=12x+54, the following table illustrates how to solve a quadratic equation, step by step by completing the square. If we could get two square terms on two sides of the quality sign, we will again get a linear equation. In those cases, we can use the other methods as discussed below.īrowse more Topics under Quadratic Equationsĭownload NCERT Solutions for Class 10 Mathematics Completing the Square MethodĮach quadratic equation has a square term. This method is convenient but is not applicable to every equation. Solving these equations for x gives: x=-4 or x=1. Thus we have either (x+4) = 0 or (x-1) = 0 or both are = 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 can factorise the terms as: (x+4)(x-1) = 0. Hence, we write x 2 + 3x – 4 = 0 as x 2 + 4x – x – 4 = 0. Consider (+4) and (-1) as the factors, whose multiplication is -4 and sum is 3. We do it such that the product of the new coefficients equals the product of a and c. Next, the middle term is split into two terms. Solution: This method is also known as splitting the middle term method. Examples of FactorizationĮxample 1: Solve the equation: x 2 + 3x – 4 = 0 Let’s see an example and we will get to know more about it.
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Hence, from these equations, we get the value of x.
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These factors, if done correctly will give two linear equations in x. Certain quadratic equations can be factorised. Robin Johnson solves the quadratic equations $3x^2-2x-1=0$ by factorisation and $3x^2-4x-2=0$ using the quadratic formula.The first and simplest method of solving quadratic equations is the factorization method. This is a not only a reliable method for finding the solutions, if they exist, but also yields an easy way of finding the associated quadratic curve's maximum or minimum point.
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You can also use the discriminant to determine how many real roots an equation has. The quadratic formula will always produce the roots of the equation in the same number of operations, but it's often quicker to use the other methods. Note that it is not always possible to factorise a quadratic expression. We see if a quadratic $q(x)$ can be factorised as $(x+r)(x+s)$ by algebraic manipulation, then the solutions of $q(x)=0$ are $x=-r,\ x=-s$. There are three commonly-used methods of solving quadratic equations: Factorising Expressions These values of $x$ are also called the roots of the equation. “Solving” this equation means finding values of $x$ which satisfy the equation. Contents Toggle Main Menu 1 Quadratic Equations 1.1 Factorising Expressions 1.2 The Quadratic Formula 1.3 Completing the Square 2 Video Example 3 Workbook 4 Test Yourself 5 See Also 6 External Resources Quadratic EquationsĪ quadratic equation is an equation of the form \ where $a\neq 0$.