Finding the Value of 'm' for Quadratic Equations with Related Roots

The problem asks for the value(s) of \(m\) such that one root of the first quadratic equation, \(x^2 - 3x + 2m = 0\), is double a root of the second quadratic equation, \(x^2 - x + m = 0\).

Setting up the Equations based on Roots

Let the roots of the second equation, \(x^2 - x + m = 0\), be \(\alpha\) and \(\beta\). From Vieta's formulas, we know that for \(ax^2 + bx + c = 0\):

• Sum of roots: \(\alpha + \beta = -b/a\)
• Product of roots: \(\alpha \beta = c/a\)

For \(x^2 - x + m = 0\):

• Sum of roots: \(\alpha + \beta = -(-1)/1 = 1\)
• Product of roots: \(\alpha \beta = m/1 = m\)

If \(\alpha\) is a root of \(x^2 - x + m = 0\), then it must satisfy the equation:

\(\alpha^2 - \alpha + m = 0 \quad (1)\)

The problem states that one root of the first equation, \(x^2 - 3x + 2m = 0\), is double a root of the second equation. Let this root of the first equation be \(2\alpha\), where \(\alpha\) is a root of the second equation.

Since \(2\alpha\) is a root of \(x^2 - 3x + 2m = 0\), it must satisfy the equation:

\((2\alpha)^2 - 3(2\alpha) + 2m = 0\)

\(4\alpha^2 - 6\alpha + 2m = 0 \quad (2)\)

Solving the System of Equations

We now have a system of two equations with \(\alpha\) and \(m\):

1. \(\alpha^2 - \alpha + m = 0\)
2. \(4\alpha^2 - 6\alpha + 2m = 0\)

From equation (1), we can express \(m\) in terms of \(\alpha\):

\(m = \alpha - \alpha^2\)

Substitute this expression for \(m\) into equation (2):

\(4\alpha^2 - 6\alpha + 2(\alpha - \alpha^2) = 0\)

\(4\alpha^2 - 6\alpha + 2\alpha - 2\alpha^2 = 0\)

Combine like terms:

\((4\alpha^2 - 2\alpha^2) + (-6\alpha + 2\alpha) = 0\)

\(2\alpha^2 - 4\alpha = 0\)

Factor out \(2\alpha\):

\(2\alpha(\alpha - 2) = 0\)

This equation gives two possible values for \(\alpha\):

• \(2\alpha = 0 \implies \alpha = 0\)
• \(\alpha - 2 = 0 \implies \alpha = 2\)

Finding the Values of m

Now, we use the expression \(m = \alpha - \alpha^2\) to find the corresponding values of \(m\) for each value of \(\alpha\).

Case 1: \(\alpha = 0\)

\(m = 0 - 0^2 = 0 - 0 = 0\)

Case 2: \(\alpha = 2\)

\(m = 2 - 2^2 = 2 - 4 = -2\)

Thus, the possible values for \(m\) are \(0\) and \(-2\).

Verification

Let's check if these values of \(m\) satisfy the original condition.

If \(m = 0\):

• Equation 1: \(x^2 - 3x + 2(0) = 0 \implies x^2 - 3x = 0 \implies x(x-3) = 0\). Roots are \(0\) and \(3\).
• Equation 2: \(x^2 - x + 0 = 0 \implies x^2 - x = 0 \implies x(x-1) = 0\). Roots are \(0\) and \(1\).

Is a root of Equation 1 double a root of Equation 2? Yes, the root \(0\) of Equation 1 is double the root \(0\) of Equation 2 (\(0 = 2 \times 0\)). This value \(m=0\) is valid.

If \(m = -2\):

• Equation 1: \(x^2 - 3x + 2(-2) = 0 \implies x^2 - 3x - 4 = 0\). Factoring: \((x-4)(x+1) = 0\). Roots are \(4\) and \(-1\).
• Equation 2: \(x^2 - x + (-2) = 0 \implies x^2 - x - 2 = 0\). Factoring: \((x-2)(x+1) = 0\). Roots are \(2\) and \(-1\).

Is a root of Equation 1 double a root of Equation 2? Yes, the root \(4\) of Equation 1 is double the root \(2\) of Equation 2 (\(4 = 2 \times 2\)). This value \(m=-2\) is valid.

Both values \(m=0\) and \(m=-2\) satisfy the given condition.

Conclusion on the Value of m

The values of \(m\) for which one root of \(x^2 - 3x + 2m = 0\) is double a root of \(x^2 - x + m = 0\) are \(0\) and \(-2\).

Revision Table: Key Concepts

Additional Information on Roots of Quadratic Equations

The roots of a quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The term \(b^2 - 4ac\) is called the discriminant (\(\Delta\)). The nature of the roots depends on the discriminant:

• If \(\Delta > 0\), there are two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root (a repeated root).
• If \(\Delta < 0\), there are two distinct complex roots.

In this problem, we did not explicitly use the quadratic formula for the roots directly, but rather the property that a root satisfies its equation and the relationships between roots based on the problem statement.