35+ CAT Quadratic Equations Questions With video Solutions

Quadratic Equations is one of the key topics in the CAT Quant Section (Algebra). The weightage for quadratic equations questions is lower. But these questions will help you boost your score in quant. It is advised to solve the questions that previously appeared in the CAT. To help the aspirants find the quadratic equations questions, we have compiled all the questions that appeared in the previous CAT papers and detailed video solutions explained by CAT experts. CAT Quadratic Equation questions appear in the CAT and other MBA entrance exams every year.  Keep practising free CAT mocks where you'll get a fair idea of how questions are asked, and type of questions asked of CAT Quadratic Equation Questions. These are a good source for practice; If you want to practice these questions, You can also download the PDF that contains all these questions with video solutions. And the best part is you can download the PDF for free without signing up

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CAT Quadratic Equations Formulas PDF

Quadratic equations are an essential topic in the quantitative aptitude section, and it is vital to have a clear understanding of the formulas related to it. To help the aspirants to ace this topic, we have made a PDF containing a comprehensive list of formulas, tips, and tricks that you can use to solve quadratic equation problems with ease and speed. Click on the below link to download CAT Quadratic Equations Formulas PDF.

1. Quadratic Equation - Given Roots.

Finding a quadratic equation:

If roots are given : (x-a)(x-b)=0 => $$x^2 - (a+b)x + ab = 0$$

If sum s and product p of roots are given: $$x^2 - sx + p = 0$$

If roots are reciprocals of roots of equation $$ax^2 + bx + c = 0$$, then equation is $$cx^2 + bx + a = 0$$

If roots are k more than roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y-k)^2 + b(y-k) + c = 0$$

If roots are k times roots of $$ax^2 + bx + c = 0$$ then equation is $$a(y/k)^2 + b(y/k) + c = 0$$

2. Quadratic Roots Formulas

The General Quadratic equation will be in the form of a$$x^{2}$$+b$$x$$+c = 0

The values of ‘x’ satisfying the equation are called the roots of the equation.

The value of roots, p and q = $$\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$

The sum of the roots = p+q = $$\dfrac{-b}{a}$$

Product of roots = p*q = $$\dfrac{c}{a}$$

If c and a are equal then the roots are reciprocal to each other.

If b = 0, then the roots are equal and are opposite in sign.

3. Discriminant Formulas

Let D denote the discriminant $$b^{2}-4ac$$. Hence, depending on the sign and value of D, nature of the roots would be as follows:

D<0 and abs(D) is not a perfect square: Roots are complex and irrational. They can be represented as p+iq and p-iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational.

D < 0 and abs(D) is a perfect square: Roots are complex but rational. They can be represented as p+iq and p-iq where p and q are both rational.

D=0 : Roots are real and equal. X = -b/2a

D>0 and D is not a perfect square: Roots are conjugate surds

D>0 and D is a perfect square: Roots are real, rational and unequal

    CAT 2023 Quadratic Equations questions

    Question 1

    The sum of all possible values of x satisfying the equation $$2^{4x^{2}}-2^{2x^{2}+x+16}+2^{2x+30}=0$$, is

    Question 2

    The equation $$x^{3} + (2r + 1)x^{2} + (4r - 1)x + 2 =0$$ has -2 as one of the roots. If the other two roots are real, then the minimum possible non-negative integer value of r is

    Question 3

    A quadratic equation $$x^2 + bx + c = 0$$ has two real roots. If the difference between the reciprocals of the roots is $$\frac{1}{3}$$, and the sum of the reciprocals of the squares of the roots is $$\frac{5}{9}$$, then the largest possible value of $$(b + c)$$ is

    Question 4

    Let $$\alpha$$ and $$\beta$$ be the two distinct roots of the equation $$2x^{2} - 6x + k = 0$$, such that ( $$\alpha + \beta$$) and $$\alpha \beta$$ are the distinct roots of the equation $$x^{2} + px + p = 0$$. Then, the value of 8(k - p) is

    Question 5

    Let k be the largest integer such that the equation $$(x-1)^{2}+2kx+11=0$$ has no real roots. If y is a positive real number, then the least possible value of $$\frac{k}{4y}+9y$$ is

    CAT 2022 Quadratic Equations questions

    Question 1

    Suppose k is any integer such that the equation $$2x^{2}+kx+5=0$$ has no real roots and the equation $$x^{2}+(k-5)x+1=0$$ has two distinct real roots for x. Then, the number of possible values of k is

    Question 2

    If $$(3+2\sqrt{2})$$ is a root of the equation $$ax^{2}+bx+c=0$$ and $$(4+2\sqrt{3})$$ is a root of the equation $$ay^{2}+my+n=0$$ where a, b, c, m and n are integers, then the value of $$(\frac{b}{m}+\frac{c-2b}{n})$$ is

    Question 3

    Let r and c be real numbers. If r and -r are roots of $$5x^{3} + cx^{2} - 10x + 9 = 0$$, then c equals

    Question 4

    Let a, b, c be non-zero real numbers such that $$b^2 < 4ac$$, and $$f(x) = ax^2 + bx + c$$. If the set S consists of all integers m such that f(m) < 0, then the set S must necessarily be

    Question 5

    The minimum possible value of $$\frac{x^{2} - 6x + 10}{3-x}$$, for $$x < 3$$, is

    CAT 2021 Quadratic Equations questions

    Question 1

    Suppose one of the roots of the equation $$ax^{2}-bx+c=0$$ is $$2+\sqrt{3}$$, Where a,b and c are rational numbers and $$a\neq0$$. If $$b=c^{3}$$ then $$\mid a\mid$$ equals.

    Question 2

    For all real values of x, the range of the function $$f(x)=\frac{x^{2}+2x+4}{2x^{2}+4x+9}$$ is:

    Question 3

    Suppose hospital A admitted 21 less Covid infected patients than hospital B, and all eventually recovered. The sum of recovery days for patients in hospitals A and B were 200 and 152, respectively. If the average recovery days for patients admitted in hospital A was 3 more than the average in hospital B then the number admitted in hospital A was

    Question 4

    Consider the pair of equations: $$x^{2}-xy-x=22$$ and $$y^{2}-xy+y=34$$. If $$x>y$$, then $$x-y$$ equals

    Question 5

    If r is a constant such that $$\mid x^2 - 4x - 13 \mid = r$$ has exactly three distinct real roots, then the value of r is

    CAT 2020 Quadratic Equations questions

    Question 1

    Let m and n be positive integers, If $$x^{2}+mx+2n=0$$ and $$x^{2}+2nx+m=0$$ have real roots, then the smallest possible value of $$m+n$$ is

    Question 2

    How many disticnt positive integer-valued solutions exist to the equation $$(x^{2}-7x+11)^{(x^{2}-13x+42)}=1$$ ?

    Question 3

    The number of distinct real roots of the equation $$(x+\frac{1}{x})^{2}-3(x+\frac{1}{x})+2=0$$ equals

    Question 4

    The number of integers that satisfy the equality $$(x^{2}-5x+7)^{x+1}=1$$ is

    CAT 2019 Quadratic Equations questions

    Question 1

    The product of the distinct roots of $$\mid x^2 - x - 6 \mid = x + 2$$ is

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