Understanding the Quadratic Equation and Roots
The problem presents a quadratic equation in the form \(ax^2+bx+c=0\), specifically \(2x^2-5px+2p^2 = 0\). The core task is to determine the value of the coefficient \(p\), given a specific condition about the equation's roots: one root is 4 units larger than the other.
Analyzing Roots Properties Using Vieta's Formulas
For any quadratic equation \(ax^2+bx+c=0\), let its roots be \(\alpha\) and \(\beta\). Vieta's formulas provide a direct link between the coefficients and the roots:
- Sum of the roots: \(\alpha + \beta = -b/a\)
- Product of the roots: \(\alpha \beta = c/a\)
In our specific equation, \(2x^2-5px+2p^2 = 0\), we identify the coefficients:
- \(a=2\)
- \(b=-5p\)
- \(c=2p^2\)
Applying Vieta's formulas to this equation yields:
- Sum of roots: \(\alpha + \beta = -(-5p)/2 = 5p/2\)
- Product of roots: \(\alpha \beta = (2p^2)/2 = p^2\)
Calculating the Value of p Step-by-Step
The problem states that one root exceeds the other by 4. We can express this relationship mathematically. Let \(\alpha\) be the larger root and \(\beta\) be the smaller root:
\(\alpha = \beta + 4\)
This is equivalent to:
\(\alpha - \beta = 4\)
Now we have a system of equations involving the roots \(\alpha\) and \(\beta\) and the unknown \(p\):
- \(\alpha + \beta = 5p/2\) (from the sum of roots)
- \(\alpha - \beta = 4\) (from the given condition)
We can solve this system to find expressions for \(\alpha\) and \(\beta\) in terms of \(p\). Add equation (1) and equation (2):
\((\alpha + \beta) + (\alpha - \beta) = 5p/2 + 4\)
\(2\alpha = 5p/2 + 4\)
Divide by 2 to find \(\alpha\):
\(\alpha = \frac{1}{2} \left( \frac{5p}{2} + 4 \right) = \frac{5p}{4} + 2\)
Next, subtract equation (2) from equation (1):
\((\alpha + \beta) - (\alpha - \beta) = 5p/2 - 4\)
\(2\beta = 5p/2 - 4\)
Divide by 2 to find \(\beta\):
\(\beta = \frac{1}{2} \left( \frac{5p}{2} - 4 \right) = \frac{5p}{4} - 2\)
Now, substitute these expressions for \(\alpha\) and \(\beta\) into the equation for the product of roots (\(\alpha \beta = p^2\)):
\(\left( \frac{5p}{4} + 2 \right) \left( \frac{5p}{4} - 2 \right) = p^2\)
The left side of the equation is in the form of a difference of squares, \((x+y)(x-y) = x^2 - y^2\), where \(x = 5p/4\) and \(y = 2\). Applying this:
\(\left( \frac{5p}{4} \right)^2 - 2^2 = p^2\)
\(\frac{25p^2}{16} - 4 = p^2\)
To solve for \(p\), rearrange the terms to isolate \(p^2\):
\(\frac{25p^2}{16} - p^2 = 4\)
Combine the \(p^2\) terms using a common denominator:
\(\frac{25p^2}{16} - \frac{16p^2}{16} = 4\)
\(\frac{9p^2}{16} = 4\)
Now, solve for \(p^2\):
\(p^2 = 4 \times \frac{16}{9}\)
\(p^2 = \frac{64}{9}\)
Finally, take the square root of both sides to find the value(s) of \(p\):
\(p = \pm \sqrt{\frac{64}{9}}\)
\(p = \pm \frac{8}{3}\)
The equation yields two possible values for \(p\): \(8/3\) and \(-8/3\). One of these values is \(8/3\).
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?
- If the sum and product of the roots of a quadratic equation are 2 and -100 respectively, then which one of the following is correct?
- 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 \(k\)?
- 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:
