If { \alpha$}$ And { \beta$}$ Are Roots Of { P(x) = Kx^2 - 30x + 45$}$ And { \alpha + \beta = 2\beta$}$, Find The Value Of { A$}$.

If $\alpha$ and $\beta$ are Roots of $P(x) = kx^2 - 30x + 45$ and $\alpha + \beta = 2\beta$, Find the Value of $A$

In this article, we will explore the relationship between the roots of a quadratic equation and the coefficients of the equation. We will use the given information to find the value of $A$, which is a coefficient of the quadratic equation.

The given quadratic equation is $P(x) = kx^2 - 30x + 45$. This equation has two roots, $\alpha$ and $\beta$. We are also given that $\alpha + \beta = 2\beta$.

The Relationship Between Roots and Coefficients

In a quadratic equation of the form $ax^2 + bx + c = 0$, the sum of the roots is given by $-\frac{b}{a}$ and the product of the roots is given by $\frac{c}{a}$. In this case, we have:

$$\alpha + \beta = -\frac{-30}{k} = \frac{30}{k}$$

We are also given that $\alpha + \beta = 2\beta$. Substituting this into the previous equation, we get:

$$2\beta = \frac{30}{k}$$

Finding the Value of $k$

We can rearrange the previous equation to solve for $k$:

$$k = \frac{30}{2\beta} = \frac{15}{\beta}$$

Finding the Value of $A$

We are given that $\alpha + \beta = 2\beta$. We can substitute this into the equation for the sum of the roots:

$$2\beta = \frac{30}{k}$$

Substituting the expression for $k$ in terms of $\beta$, we get:

$$2\beta = \frac{30}{\frac{15}{\beta}} = \frac{30\beta}{15} = 2\beta$$

This equation is true for all values of $\beta$. Therefore, we can conclude that:

$$A = k = \frac{15}{\beta}$$

However, we are not given the value of $\beta$. To find the value of $A$, we need to use the fact that $\alpha$ and $\beta$ are roots of the quadratic equation.

Using Vieta's Formulas

Vieta's formulas state that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, and the product of the roots is equal to the constant term. In this case, we have:

$$\alpha + \beta = -\frac{-30}{k} = \frac{30}{k}$$

$$\alpha\beta = \frac{45}{k}$$

We are given that $\alpha + \beta = 2\beta$. Substituting this into the previous equation, we get:

$$2\beta = \frac{30}{k}$$

Substituting the expression for $k$ in terms of $\beta$, we get:

$$2\beta = \frac{30}{\frac{15}{\beta}} = \frac{30\beta}{15} = 2\beta$$

This equation is true for all values of $\beta$. Therefore, we can conclude that:

$$A = k = \frac{15}{\beta}$$

However, we are not given the value of $\beta$. To find the value of $A$, we need to use the fact that $\alpha$ and $\beta$ are roots of the quadratic equation.

Finding the Value of $\beta$

We can use the fact that $\alpha$ and $\beta$ are roots of the quadratic equation to find the value of $\beta$. We have:

$$\alpha\beta = \frac{45}{k}$$

Substituting the expression for $k$ in terms of $\beta$, we get:

$$\alpha\beta = \frac{45}{\frac{15}{\beta}} = \frac{45\beta}{15} = 3\beta$$

We are given that $\alpha + \beta = 2\beta$. Substituting this into the previous equation, we get:

$$3\beta = \alpha(2\beta)$$

Substituting the expression for $\alpha$ in terms of $\beta$, we get:

$$3\beta = (2\beta - \beta)(\beta)$$

Simplifying, we get:

$$3\beta = \beta^2$$

This equation is true for all values of $\beta$. Therefore, we can conclude that:

$$\beta = 3$$

Finding the Value of $A$

Now that we have found the value of $\beta$, we can find the value of $A$. We have:

$$A = k = \frac{15}{\beta}$$

Substituting the value of $\beta$, we get:

$$A = k = \frac{15}{3} = 5$$

Therefore, the value of $A$ is 5.

In this article, we have used the given information to find the value of $A$, which is a coefficient of the quadratic equation. We have used the fact that $\alpha$ and $\beta$ are roots of the quadratic equation, and the relationship between the roots and the coefficients of the equation. We have also used Vieta's formulas to find the value of $\beta$, and then used this value to find the value of $A$. The final answer is 5.
Q&A: If $\alpha$ and $\beta$ are Roots of $P(x) = kx^2 - 30x + 45$ and $\alpha + \beta = 2\beta$, Find the Value of $A$

A: The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, and the product of the roots is equal to the constant term. In this case, we have:

$$\alpha + \beta = -\frac{-30}{k} = \frac{30}{k}$$

$$\alpha\beta = \frac{45}{k}$$

$$$$

A: We can rearrange the equation for the sum of the roots to solve for $k$:

$$k = \frac{30}{2\beta} = \frac{15}{\beta}$$

$$

A: We can use the fact that $\alpha$ and $\beta$ are roots of the quadratic equation to find the value of $\beta$. We have:

$$\alpha\beta = \frac{45}{k}$$

Substituting the expression for $k$ in terms of $\beta$, we get:

$$\alpha\beta = \frac{45}{\frac{15}{\beta}} = \frac{45\beta}{15} = 3\beta$$

We are given that $\alpha + \beta = 2\beta$. Substituting this into the previous equation, we get:

$$3\beta = \alpha(2\beta)$$

Substituting the expression for $\alpha$ in terms of $\beta$, we get:

$$3\beta = (2\beta - \beta)(\beta)$$

Simplifying, we get:

$$3\beta = \beta^2$$

This equation is true for all values of $\beta$. Therefore, we can conclude that:

$$\beta = 3$$

$$

A: Now that we have found the value of $\beta$, we can find the value of $A$. We have:

$$A = k = \frac{15}{\beta}$$

Substituting the value of $\beta$, we get:

$$A = k = \frac{15}{3} = 5$$

Therefore, the value of $A$ is 5.

A: Vieta's formulas state that the sum of the roots of a quadratic equation is equal to the negation of the coefficient of the linear term, and the product of the roots is equal to the constant term. In this case, we have used Vieta's formulas to find the value of $\beta$, and then used this value to find the value of $A$.

$$

A: Yes, we can find the value of $A$ without using Vieta's formulas. We can use the fact that $\alpha$ and $\beta$ are roots of the quadratic equation to find the value of $\beta$, and then use this value to find the value of $A$. However, using Vieta's formulas makes the problem much easier to solve.

A: The final answer to the problem is 5.