If the roots of the quadratic equation x 2 - 4x - log 10 N = 0 are real, then what is the minimum value of N ?
Understanding the Problem: Finding Minimum N for Real Roots
The question asks us to find the minimum value of \(N\) such that the quadratic equation \(x^2 - 4x - \log_{10} N = 0\) has real roots. For a quadratic equation of the form \(ax^2 + bx + c = 0\) to have real roots, its discriminant (\(\Delta\)) must be greater than or equal to zero (\(\Delta \ge 0\)).
In the given equation, \(x^2 - 4x - \log_{10} N = 0\), we can identify the coefficients:
- \(a = 1\)
- \(b = -4\)
- \(c = -\log_{10} N\)
We also need to remember that for \(\log_{10} N\) to be defined, the value of \(N\) must be positive, i.e., \(N > 0\).
Calculating the Discriminant for Real Roots
The discriminant \(\Delta\) of a quadratic equation is calculated using the formula: \(\Delta = b^2 - 4ac\).
Let's substitute the values of \(a\), \(b\), and \(c\) from our equation into the discriminant formula:
\(\Delta = (-4)^2 - 4(1)(-\log_{10} N)\)
Now, let's simplify the expression:
\(\Delta = 16 - 4(-\log_{10} N)\)
\(\Delta = 16 + 4 \log_{10} N\)
Condition for Real Roots
For the quadratic equation to have real roots, the discriminant must be non-negative:
\(\Delta \ge 0\)
Substitute the expression for \(\Delta\):
\(16 + 4 \log_{10} N \ge 0\)
Solving the Inequality for N
Now we need to solve this inequality to find the possible values of \(N\). First, isolate the logarithm term:
\(4 \log_{10} N \ge -16\)
Divide both sides by 4:
\(\log_{10} N \ge \frac{-16}{4}\)
\(\log_{10} N \ge -4\)
To solve for \(N\), we use the definition of logarithms. The expression \(\log_b y = x\) is equivalent to \(y = b^x\). In our case, the base \(b\) is 10, the 'answer' \(x\) is -4, and the number \(y\) is \(N\).
So, applying this rule:
\(N \ge 10^{-4}\)
We can calculate the value of \(10^{-4}\):
\(10^{-4} = \frac{1}{10^4} = \frac{1}{10 \times 10 \times 10 \times 10} = \frac{1}{10000}\)
So the inequality for \(N\) is:
\(N \ge \frac{1}{10000}\)
Finding the Minimum Value of N
The inequality \(N \ge \frac{1}{10000}\) tells us that \(N\) can be \(\frac{1}{10000}\) or any value greater than \(\frac{1}{10000}\).
The smallest possible value that \(N\) can take while satisfying this condition is \(\frac{1}{10000}\).
We must also remember the initial constraint that \(N > 0\) for \(\log_{10} N\) to be defined. Since \(\frac{1}{10000}\) is positive, the condition \(N \ge \frac{1}{10000}\) automatically satisfies \(N > 0\).
Therefore, the minimum value of \(N\) for which the quadratic equation has real roots is \(\frac{1}{10000}\).
Revision Table: Quadratic Equation Real Roots
- Problem Type: Finding parameter value for real roots
- Key Concept: Discriminant (\(\Delta\)) of a quadratic equation
- Condition for Real Roots: \(\Delta \ge 0\)
- Quadratic Equation Form: \(ax^2 + bx + c = 0\)
- Discriminant Formula: \(\Delta = b^2 - 4ac\)
- Logarithm Definition: If \(\log_b y = x\), then \(y = b^x\)
Additional Information: Discriminant and Nature of Roots
The discriminant \(\Delta = b^2 - 4ac\) of a quadratic equation \(ax^2 + bx + c = 0\) is a crucial value that determines the nature of its roots.
- If \(\Delta > 0\): The equation has two distinct real roots.
- If \(\Delta = 0\): The equation has exactly one real root (a repeated or double root).
- If \(\Delta < 0\): The equation has two distinct complex (non-real) roots.
In our problem, the requirement for "real roots" means the roots must be either distinct real roots (\(\Delta > 0\)) or a single real root (\(\Delta = 0\)). Combining these conditions gives \(\Delta \ge 0\), which is what we used to solve the problem involving the minimum value of \(N\).
Understanding the discriminant is fundamental for analyzing quadratic equations in algebra and various mathematical applications.
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