Finding Roots of the First Quadratic Equation
Given the quadratic equation: \(x^2 - 2bx + c^2 = 0\)
Let the roots be \(\alpha\) and \(\beta\). Using Vieta's formulas:
- Sum of roots: \(\alpha + \beta = -(\frac{-2b}{1}) = 2b\)
- Product of roots: \(\alpha \beta = \frac{c^2}{1} = c^2\)
Calculating Arithmetic Mean (A) and Geometric Mean (G)
The arithmetic mean (A) of \(\alpha\) and \(\beta\) is: \(A = \frac{\alpha + \beta}{2} = \frac{2b}{2} = b\)
The geometric mean (G) of \(\alpha\) and \(\beta\) is: \(G = \sqrt{\alpha \beta}\) Since \(\alpha \beta = c^2\) and \(c\) is a positive real number, \(G = \sqrt{c^2} = c\).
Finding Roots of the Second Quadratic Equation
The second quadratic equation is: \(x^2 - (b+c)x + bc = 0\)
This equation can be factored: We look for two numbers that multiply to \(bc\) and add to \(b+c\). These numbers are \(b\) and \(c\). So, the equation becomes: \((x - b)(x - c) = 0\)
The roots of this equation are \(x = b\) and \(x = c\).
Relating Roots to A and G
We found that \(A = b\) and \(G = c\). The roots of the second equation are \(b\) and \(c\). Therefore, the roots of the equation \(x^2 - (b+c)x + bc = 0\) are A and G.
Similar Questions
If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?
If the roots of the equation x 2+ px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?
If x 2- px + 4 > 0 for all real values of x, then which one of the following is correct?
How many real roots does the equation x 2+ 3|x| + 2 = 0 have?
What is the GM of the roots of the equation ?
In solving a problem that reduces to a quadratic equation, one student makes a mistake in the constant term and obtains 8 and 2 for roots. Another student makes a mistake only in the coefficient of first-degree term and finds -9 and -1 for roots.
The correct equation is
Suppose f(x) is such a quadratic expression that it is positive for all real x.
If g(x) = f(x) + f'(x) + f”(x), then for any real xIf α and β are the roots of x 2+ x + 1 = 0, then what is \(\mathop \sum \limits_{j = 0}^3 \left( {{\alpha ^j} + {\beta ^j}} \right)\) equal to?
If the sum of the roots of the equation ax 2+ bx + c = 0 is equal to the sum of their squares then
Under which one of the following condition will the quadratic equation x 2+ mx + 2 = 0 always have real roots?
Important Questions from Quadratic Equations
If the graph of a quadratic polynomial lies entirely above x-axis, then which one of the following is correct?
If the roots of the equation x 2+ px + q = 0 are tan 19° and tan 26°, then which one of the following is correct?
If x 2- px + 4 > 0 for all real values of x, then which one of the following is correct?
If 2 + i is a root of the equation x 2- ax + 1 = 0, then the value of a is:
If \(\tan \left( {\frac{α }{2}} \right)\) and \(\tan \left( {\frac{β }{2}} \right)\) are the roots of the equation 8x 2- 26x + 15 = 0, then the value of cos (α + β) will be
