Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

GMAT: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

The document Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT is a part of the GMAT Course Quantitative Aptitude for GMAT.
All you need of GMAT at this link: GMAT

Section - 1
Distribute the following expressions.
Ques 1: (x+ 2)(x- 3)
Ans:
(x + 2 )(x -3 )= x2- 3x + 2x - 6 = x2- x - 6

Ques 2: (2s+1)(s + 5)
Ans:
(2s+ 1)(s+5) = 2s2 + 10s + s + 5 = 2s2+ 11s + 5

Ques 3: (5 + a)(3 + a)
Ans:
(5 + a)(3 + a) = 15 + 5a + 3a + a2 = a2 + 8 a + 15

Ques 4: (3 - z)(z + 4)
Ans:
(3 - z)(z + 4) = 3z + 12 - z2 - 4z = -z2 - z + 12

Section - 2
Ques 5: x2 - 2x = 0
Ans: 
x2 — 2x = 0
x(x — 2) = 0
x= 0
OR (x -2 ) = 0 -> x = 2

Ques 6: z2 = -5 z
Ans: 

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 7: y2 + 4y + 3 = 0
Ans:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 8: y2- 11y + 30 = 0
Ans:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 9: y2 + 3 y = 0
Ans:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 10: y2 + 12y + 36 = 0
Ans:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Section - 3
Simplify the following expressions.
Ques 11: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:
  a + b1 The key to simplifying this expression is to recognize the special product:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
After replacing the original numerator with (a + b)(a - b), we can cancel the (a - b) in the numerator with the (a - b) in the denominator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 12: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 13: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: 
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Because we have a common factor in both terms of the numerator, we can divide that factor out in order to simplify further. This is often a useful move when we are asked to add or subtract exponents with the same base:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 14: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: It is tempting to expand the quadratic term in the numerator, but we should try to simplify first. Notice that none of the answer choices are fractions. Therefore, we need to look for a way to cancel (2t — 1) from the denominator. To make this task easier, enclose every (2t — 1) term in parentheses and then simplify. We can factor (2t - 1) out of the numerator:

Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Alternatively, we can introduce a new variable x defined as x = 2t — 1. In general, it is not a good idea to introduce more variables than strictly necessary, but in this case the new variable can make it easier to see how the math works:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
We can then simplify the expression in terms of x:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
And we then finish by replacing x with 2t - 1
x + 1 = (2t — 1) + 1 = 2t

Section - 4
Simplify the following expressions.
Ques 15: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT 
(A) 5 - x (B) x - 5 (C) x + 5
Ans: 
We can make this problem a lot simpler if we begin by factoring 3 out of both the nu-

merator and the denominator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
The answers are not fractions, so well have to get rid of the denominator. Factor the numerator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 16: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans:
This might seem nearly impossible to factor down until we take out the common term: ab. Then, we are left with one of our familiar special products in the denominator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

Ques 17: Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Ans: 
There's no obvious way to proceed through this question. The best bet is to try to

simplify before multiplying. Notice that x3 can be factored out of the numerator and x2 can be factored out of the denominator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
Notice now that we can cancel x2, from the fraction. Also, we have (x2 - 1) in the numerator, which we can factor:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT
We don’t have a match with any of the answer choices yet. None of the numerators in the answer choices have parenthetical expressions. We should multiply the numerator:
Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT

The document Solved Examples - Quadratic Equations Notes | Study Quantitative Aptitude for GMAT - GMAT is a part of the GMAT Course Quantitative Aptitude for GMAT.
All you need of GMAT at this link: GMAT
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