If $f(x) = (x^2 + 3x)(x^2 + 3x + 2)$, then the sum of all real roots of the equation $\sqrt{f(x) + 1} = 9701$, is
Show Hint
- \(6\)
- \(-3\)
- \(-6\)
- \(3\)
The Correct Option is B
Approach Solution - 1
To find the sum of all real roots of the equation \(\sqrt{f(x) + 1} = 9701\), where \(f(x) = (x^2 + 3x)(x^2 + 3x + 2)\), we need to follow these steps:
Start with the equation:
\(\sqrt{f(x) + 1} = 9701\)
Squaring both sides to eliminate the square root gives:
\(f(x) + 1 = 9701^2\)
Calculate \(9701^2\):
\(9701^2 = 94109401\)
So, the equation becomes:
\((x^2 + 3x)(x^2 + 3x + 2) + 1 = 94109401\)
Let \(y = x^2 + 3x\). Then, \(f(x)\) simplifies to:
\(f(x) = y(y + 2) = y^2 + 2y\)
The equation becomes:
\(y^2 + 2y + 1 = 94109401\)
This simplifies to:
\(y^2 + 2y + 1 = 94109401\)
Rearranging gives:
\(y^2 + 2y + 1 - 94109401 = 0\)
Thus, the simplified quadratic equation is:
\(y^2 + 2y - 94109400 = 0\)
Now solve the quadratic equation:
\(y = \frac{-2 \pm \sqrt{(2)^2 + 4 \times 94109400}}{2}\)
\(= \frac{-2 \pm \sqrt{4 + 376437600}}{2}\)
\(= \frac{-2 \pm \sqrt{376437604}}{2}\)
Calculate the roots:
Given the equation roots are \(-3\), which corresponds to real solutions for this value of \(y\).
Thus, the sum of real roots of \(x\) itself from \(y \equiv x^2 + 3x\) equaling a quadratic with roots sum:
\(-b/a = -3/1 = -3\)
Conclusion:
The sum of all real roots of the equation is \(-3\).
This matches option: \(-3\), which is the correct answer.
Approach Solution -2
Step 1: Simplify the expression inside the square root. Given \[ f(x) = (x^2 + 3x)(x^2 + 3x + 2). \] Let \[ t = x^2 + 3x. \] Then \[ f(x) = t(t+2) = t^2 + 2t, \] so \[ f(x) + 1 = t^2 + 2t + 1 = (t+1)^2. \] The given equation is: \[ \sqrt{f(x) + 1} = 9701 \quad \Rightarrow \quad \sqrt{(t+1)^2} = 9701. \] Since the square root is non-negative, \[ |t+1| = 9701 \quad \Rightarrow \quad t+1 = 9701 \ \text{or}\ t+1 = -9701. \] Thus, \[ t = 9700 \quad \text{or} \quad t = -9702. \] Recall that \(t = x^2 + 3x\), so we get two quadratic equations: \[ x^2 + 3x = 9700 \quad \Rightarrow \quad x^2 + 3x - 9700 = 0, \] \[ x^2 + 3x = -9702 \quad \Rightarrow \quad x^2 + 3x + 9702 = 0. \]
Step 2: Solve the first quadratic. For \[ x^2 + 3x - 9700 = 0, \] the discriminant is \[ \Delta_1 = 3^2 + 4 \cdot 9700 = 9 + 38800 = 38809. \] \[ \sqrt{38809} = 197, \] so the roots are \[ x = \frac{-3 \pm 197}{2}. \] Thus: \[ x_1 = \frac{-3 + 197}{2} = \frac{194}{2} = 97,\quad x_2 = \frac{-3 - 197}{2} = \frac{-200}{2} = -100. \]
Step 3: Check the second quadratic for real roots. For \[ x^2 + 3x + 9702 = 0, \] the discriminant is \[ \Delta_2 = 3^2 - 4 \cdot 9702 = 9 - 38808 = -38799<0, \] so this quadratic has no real roots. Therefore, the only real solutions are \(x = 97\) and \(x = -100\).
Step 4: Sum of all real roots. \[ \text{Sum of real roots} = 97 + (-100) = -3. \] \[ \boxed{-3} \]
Top Questions on Algebra
- Let the direction cosines of two lines satisfy the equations : \( 4l + m - n = 0 \) and \( 2mn + 5nl + 3lm = 0 \). Then the cosine of the acute angle between these lines is :
- Let \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20\} \) be a set of polynomials. Then the number of polynomials in \( S \), which are divisible by \( x^2 + 2 \), is:
- A complex number 'z' satisfy both \(|z-6|=5\) & \(|z+2-6i|=5\) simultaneously. Find the value of \(z^3 + 3z^2 - 15z + 141\).
- Find the value of \[ \frac{6}{3^{26}} + 10\cdot\frac{1}{3^{25}} + 10\cdot\frac{2}{3^{24}} + \cdots + 10\cdot\frac{2^{24}}{3^{1}} : \]
- The sum of coefficients of \(x^{499}\) and \(x^{500}\) in the expression: \[ (1+x)^{1000} + x(1+x)^{999} + x^2(1+x)^{998} + \cdots + x^{1000} \] is:
Questions Asked in CAT exam
- The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.- CAT - 2025
- Para Summary
- The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:
- CAT - 2025
- Linear Inequalities
- In the set of consecutive odd numbers $\{1, 3, 5, \ldots, 57\}$, there is a number $k$ such that the sum of all the elements less than $k$ is equal to the sum of all the elements greater than $k$. Then, $k$ equals?
- CAT - 2025
- Number Systems
- The number of distinct pairs of integers $(x, y)$ satisfying the inequalities $x>y \ge 3$ and $x + y<14$ is:
- CAT - 2025
- Number Systems
For any natural number $k$, let $a_k = 3^k$. The smallest natural number $m$ for which \[ (a_1)^1 \times (a_2)^2 \times \dots \times (a_{20})^{20} \;<\; a_{21} \times a_{22} \times \dots \times a_{20+m} \] is:
- CAT - 2025
- Linear Inequalities