Identity Condition for Polynomial Equations
An equation of the form \(Ax^2 + Bx + C = 0\) is considered an identity if it holds true for all possible values of the variable \(x\). For this to be true, the coefficients of the polynomial must all be equal to zero. That is, we must have \(A = 0\), \(B = 0\), and \(C = 0\) simultaneously.
Setting Coefficients to Zero
In the given equation, \((k^2-5k+4)x^2+(k^2-3k-4)x+(k^2-4k)=0\), we identify the coefficients:
- \(A = k^2-5k+4\)
- \(B = k^2-3k-4\)
- \(C = k^2-4k\)
For the equation to be an identity, we set each coefficient to zero:
- \(k^2-5k+4 = 0\)
- \(k^2-3k-4 = 0\)
- \(k^2-4k = 0\)
Solving for k in Each Equation
We solve each quadratic equation for \(k\) to find the possible values:
Equation 1: \(k^2-5k+4 = 0\)
Factoring the quadratic expression:
\((k-1)(k-4) = 0\)
This gives us two possible values for \(k\): \(k=1\) or \(k=4\).
Equation 2: \(k^2-3k-4 = 0\)
Factoring the quadratic expression:
\((k-4)(k+1) = 0\)
This gives us two possible values for \(k\): \(k=4\) or \(k=-1\).
Equation 3: \(k^2-4k = 0\)
Factoring the expression:
\(k(k-4) = 0\)
This gives us two possible values for \(k\): \(k=0\) or \(k=4\).
Finding the Common Value of k
For the original equation to be an identity, the value of \(k\) must satisfy all three conditions simultaneously. We look for the common value among the solutions from the three equations:
- Solutions for Eq 1: \(\{1, 4\}\)
- Solutions for Eq 2: \(\{-1, 4\}\)
- Solutions for Eq 3: \(\{0, 4\}\)
The only value of \(k\) that appears in all three sets of solutions is \(k=4\).
Conclusion
Therefore, the value of \(k\) for which the given equation is an identity is 4.
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 \(k\)?
- 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 :
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:
