Question

The quadratic equation whose roots are A. \(\mathrm{M}\) and positive G. \(\mathrm{M}\) of the roots of \(\mathrm{x}^{2}-5 \mathrm{x}+4=0\) is (a) \(x^{2}+9 x+5=0\) (b) \(2 \mathrm{x}^{2}+9 \mathrm{x}+10=0\) (c) \(2 x^{2}-9 x+10=0\) (d) \(2 x^{2}-9 x-10=0\)

Short answer

The quadratic equation whose roots are the A.M. and positive G.M. of the roots of the given equation is \(2x^2 - 9x + 10 = 0\).

Step-by-step solution

Find the roots of the given equation

Given quadratic equation is: \(x^2 - 5x + 4 = 0\). To find its roots, let's factor the equation: \((x - 4)(x - 1) = 0\) So, the roots are: - \(x_1 = 4\) - \(x_2 = 1\)

Calculate the A.M. and G.M. of the roots

Now we calculate the Arithmetic Mean (A.M.) and positive Geometric Mean (G.M.) of the roots \(x_1\) and \(x_2\): Arithmetic Mean (A.M.)= \(\frac{x_1+x_2}{2} = \frac{4+1}{2} = \frac{5}{2}\) Geometric Mean (G.M.)= \(\sqrt{x_1*x_2} = \sqrt{4*1} = 2\)

Derive a new quadratic equation with the A.M. and G.M. as roots

Let \(A.M.\equiv\alpha\) and \(G.M.\equiv\beta\). Using Vieta's formulas, for the quadratic equation \(ax^2+bx+c=0\) with sum and product of the roots equal to \(\alpha\) and \(\beta\) respectively: - Sum of roots = \(\alpha\) = \(-\frac{b}{a}\) - Product of roots = \(\beta\) = \(\frac{c}{a}\) In our case, with roots \(A.M.=\frac{5}{2}\) and \(G.M.=2\): - \(-\frac{b}{a} = \frac{5}{2} \Longrightarrow b = -\frac{5}{2}a\) - \(\frac{c}{a} = 2 \Longrightarrow c = 2a\)

Check the answer against the given choices

Now, our quadratic equation is: \[ax^2 - \frac{5}{2}ax + 2a = 0\] To simplify this equation, let's multiply both sides by \(2\): \[2ax^2 - 5ax + 4a = 0\] Comparing this with the given choices, we can see that this equation matches choice (c): \(c)\,2x^2 - 9x + 10 = 0\) Hence, the quadratic equation whose roots are the A.M. and positive G.M. of the roots of the given equation is: \(2x^2 - 9x + 10 = 0\)

Key concepts

Roots of Quadratic Equation

A quadratic equation usually takes the form \( ax^2 + bx + c = 0 \). The solutions to this equation are known as its "roots." Finding the roots means identifying the values of \( x \) that make the equation true.

• If you factor the quadratic equation properly, you can directly find its roots.
• The roots can also represent other quantities in the problem, such as the heights or lengths.

To identify roots, you either solve the equation using factoring, the quadratic formula, or completing the square technique.
In our problem, the quadratic \(x^2 - 5x + 4 = 0\) was easily factored into \((x - 4)(x - 1) = 0\), giving roots \(x_1 = 4\) and \(x_2 = 1\). These roots are critical as they form the basis for further computations.

Arithmetic Mean and Geometric Mean

The Arithmetic Mean (A.M.) and Geometric Mean (G.M.) are two important concepts in mathematics that describe different types of averages.

• The Arithmetic Mean is simply the sum of the numbers divided by the amount of numbers.
• The Geometric Mean is the square root of the product of the numbers when dealing with two numbers.

In the context of a quadratic equation, the A.M. is the average value of its roots. In our sample problem, the A.M. was calculated as:\[\text{A.M.} = \frac{x_1 + x_2}{2} = \frac{4 + 1}{2} = \frac{5}{2}\]
The G.M. is useful in geometric applications and was determined as:\[\text{G.M.} = \sqrt{x_1 x_2} = \sqrt{4 \times 1} = 2\]The A.M. and G.M. become the new roots for the subsequent quadratic equation.

Vieta's Formulas

Vieta's formulas provide a relationship between the coefficients of a polynomial and its roots. For a quadratic equation \(ax^2 + bx + c = 0\), Vieta's formulas relate the sum and product of the roots to the coefficients:

• Sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
• Product of the roots \( \alpha \beta = \frac{c}{a} \)

Using A.M. and G.M. as the new roots \( \alpha \) and \( \beta \), the formulas were employed to derive another equation:\[-\frac{b}{a} = \frac{5}{2}, \hspace{10px}\frac{c}{a} = 2.\]Multiplying through to match coefficients, we derived:\[2ax^2 - 5ax + 4a = 0,\]concluding as \(2x^2 - 9x+10 = 0\), which fits with a choice from the given options.