Quadratic Equations Practice Set For All Competitive Exams
Quadratic Equations are the most astounding scoring territory in the different Exams not withstanding for non-math understudies”. As abnormal as it may appear, the announcement holds valid for some reasons.
The conditions look alarming, striking apprehension even before the competitor peruses the inquiry. Numerous competitors flee from these high scoring questions. Looking unnerving doesn’t make them extreme! Read on to know how to settle them effectively.
The key to settling these inquiries is to see that you needn’t bother with the genuine estimations of x and y to stamp the correct answer. On the off chance that you aren’t comfortable with the directions, do experience the inset. The hardest piece of this inquiry is to comprehend when to check which choice. You should ace them even before you figure out how to understand a condition.
This question can be completed without even using a paper and pen.
In any quadratic equation of the form ax^2+bx+c=0, the sum of the roots is -b/a and the product of the roots is c/a. We can quickly notice that the product of the roots is negative for both the equations. We don’t need to know the actual value. When will the product of two quantities be negative – only if one of them is positive and the other is negative. That is the hint for us to mark the answer.
When the product of the roots is negative, it means that out of the two values of x, one is positive and another is negative. Thus, in the above situation, the positive value for x will definitely be more than the negative value of y and the positive value for y will definitely be more than the negative value of x. As a result, we cannot say which the two variables are greater. Go ahead and mark the option as (E). This is generally the case with most bad equations in the exam.
Methods of Solving Quadratic Equations
There are three main methods for solving quadratic equations:
- 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.
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.
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:
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
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