Understanding Quadratic Equations and Real Roots
The problem asks us to find the greatest value of \(k\) for the quadratic equation \(2x^2 - 4x + k = 0\) such that it has real roots. A quadratic equation is generally written in the form \(ax^2 + bx + c = 0\). For an equation to have real roots, its discriminant must be non-negative (greater than or equal to zero).
Determining the Discriminant Condition
The discriminant, denoted by \(\Delta\), is calculated using the formula:
\( \Delta = b^2 - 4ac \)
For a quadratic equation to have real roots, the condition is:
\( \Delta \ge 0 \)
If \(\Delta > 0\), the equation has two distinct real roots.
If \(\Delta = 0\), the equation has exactly one real root (or two equal real roots).
If \(\Delta < 0\), the equation has no real roots (it has two complex conjugate roots).
Applying Discriminant to \(2x^2 - 4x + k = 0\)
In the given equation, \(2x^2 - 4x + k = 0\), we can identify the coefficients:
- \(a = 2\)
- \(b = -4\)
- \(c = k\)
Now, let's calculate the discriminant for this specific equation:
\( \Delta = (-4)^2 - 4(2)(k) \)
\( \Delta = 16 - 8k \)
Solving for the Greatest Value of \(k\)
To ensure the equation has real roots, we must satisfy the condition \(\Delta \ge 0\). Substitute the calculated discriminant:
\( 16 - 8k \ge 0 \)
Now, we need to solve this inequality for \(k\).
- Add \(8k\) to both sides:
\( 16 \ge 8k \) - Divide both sides by 8 (since 8 is positive, the inequality sign remains the same):
\( \frac{16}{8} \ge k \) - Simplify the fraction:
\( 2 \ge k \)
This inequality, \(2 \ge k\), means that \(k\) must be less than or equal to 2 (\(k \le 2\)).
The question asks for the greatest value of \(k\) that satisfies this condition. Since \(k\) must be less than or equal to 2, the greatest possible value for \(k\) is 2.
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?
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:
