Quadratic Equation from Sum and Product of Roots
This problem asks us to find the specific quadratic equation when we know the sum and the product of its roots.
Understanding Quadratic Equations and Roots
A standard quadratic equation has the form: \(ax^2 + bx + c = 0\), where \(a \neq 0\).
For such an equation, the relationships between the coefficients (\(a, b, c\)) and the roots (let's call them \(\alpha\) and \(\beta\)) are:
- Sum of roots: \(\alpha + \beta = -\frac{b}{a}\)
- Product of roots: \(\alpha \beta = \frac{c}{a}\)
A simpler way to write a quadratic equation, especially when the leading coefficient (\(a\)) is 1, is by using the sum and product directly:
\(x^2 - (\text{Sum of roots})x + (\text{Product of roots}) = 0\)Applying the Given Values
In this question, we are given:
- Sum of roots = 2
- Product of roots = -100
Now, we substitute these values into the formula:
\(x^2 - (2)x + (-100) = 0\)Simplifying this equation, we get:
\(x^2 - 2x - 100 = 0\)Analyzing the Options
Let's check the given options based on our derived equation:
- Option 1: "There are infinitely many such equations having different roots." This is incorrect. While we can multiply the entire equation by any non-zero constant (e.g., \(2x^2 - 4x - 200 = 0\)), the fundamental relationship defined by the sum and product leads to a unique standard form of the equation (\(x^2 + (b/a)x + c/a = 0\)). The roots themselves are also uniquely determined (up to order) by the sum and product.
- Option 2: "There is only one such equation which is \(x^2 + 2x - 100 = 0\)." If this were the equation, the sum of the roots would be \(-(2)/1 = -2\), not 2. So, this is incorrect.
- Option 3: "There is only one such equation which is \(x^2 - 2x - 100 = 0\)." This matches the equation we derived using the formula. The sum of roots is \(-(-2)/1 = 2\), and the product of roots is \(-100/1 = -100\). This matches the given information.
- Option 4: "There is no such equation." This is incorrect because we have successfully constructed the equation.
Conclusion
Based on the standard form of a quadratic equation derived from the sum and product of its roots, the correct equation is \(x^2 - 2x - 100 = 0\). This confirms that there is indeed only one such unique standard quadratic equation.
Similar Questions
- If the equation \(x \cos \theta = x^2 + p\) has a real solution for every \(\theta\) where \(0 \le \theta \le \frac{\pi}{4}\), then which one of the following is correct?
Consider the following in respect of a positive real number \(x\) :
I. \(x+\frac{1}{x} >1\)
II. \(x+\frac{1}{x} > 2\)
III. \((x+\frac{1}{x})^2 > 9\)
Which of the above are correct?
- If \(\left(\frac{a}{b}\right)x^2-\left(\frac{a+b}{b}\right)x+b = 0\), then what are the roots of this equation?
- What is the greatest value of \(k\) for which \(2x^2 - 4x + k = 0\) has real roots?
Let $p, q$ be the roots of the equation $x^2 + mx - n = 0$ and $m, n$ be the roots of the equation $x^2 + px - q = 0$ ($m, n, p, q$ are non-zero numbers). Which of the following statements is/are correct?
I. $m(m + n) = -1$
II. $p + q = 1$
Select the answer using the code given below:If the roots of the equation \(x^2-(k-2)x + (k + 1) = 0\) are equal, then what are the values of \(k\)?
- If one root of the equation \(2x^2-5px+2p^2 = 0\) exceeds the other by 4, then what is the value of \(p\)?
- Let Q = x² + bx + c. If the sum of the roots is equal to product of the roots of the equation Q = 0, then which of the following statements is/are correct ?
I. Q can be a perfect square.
II. Q is positive for all real values of x.
Select the answer using the code given below : - What is the value of \(k\) for which \((k^2-5k+4)x^2+(k^2-3k-4)x+(k^2-4k)=0\) is an identity ?
Important Questions from Quadratic equation
Find the minimum value of 2x² – 5x – 3 and also find x.
A quadratic equation $x^2 + 3x + k = 0$ has a discriminant equal to 9. What is the value of k?
- Reduce the equation $x^4 - 13x^2 + 36 = 0$ into a quadratic equation and find the roots of the reduced quadratic equation.
If both roots of the quadratic equation, $x^2 + 3x + 2 = 0$, are also roots of another quadratic equation, $ax^2 + bx + c = 0$, then which of the following must be true?
- The nature of the roots of the quadratic equation $3x^{2} – 5x + 2 = 0$ is:
