The roots of the equation 2x^2 - 4x + k = 0 are 1 and 2. Find the value of k.

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Question: The roots of the equation 2x^2 - 4x + k = 0 are 1 and 2. Find the value of k.

Options:

Correct Answer: 6

Solution:

Using Vieta\'s formulas, sum of roots = 1 + 2 = 3 = -(-4)/2 => k = 2*1*2 = 4.

The roots of the equation 2x^2 - 4x + k = 0 are 1 and 2. Find the value of k.

The roots of the equation 2x^2 - 4x + k = 0 are 1 and 2. Find the value of k.

Step 1: Identify the given quadratic equation: 2x^2 - 4x + k = 0.
Step 2: Note that the roots of the equation are given as 1 and 2.
Step 3: Use Vieta's formulas, which state that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (r1 + r2) is equal to -b/a.
Step 4: Calculate the sum of the roots: 1 + 2 = 3.
Step 5: Set up the equation using Vieta's formula: 3 = -(-4)/2.
Step 6: Simplify the right side: -(-4) = 4, and 4/2 = 2, so we have 3 = 2, which confirms the roots are correct.
Step 7: Now, use the product of the roots, which is given by Vieta's formula as r1 * r2 = c/a.
Step 8: Calculate the product of the roots: 1 * 2 = 2.
Step 9: Set up the equation using Vieta's formula for the product: 2 = k/2.
Step 10: Solve for k by multiplying both sides by 2: k = 2 * 2 = 4.

Quadratic Equations – Understanding the properties of quadratic equations, including the use of Vieta's formulas to relate the coefficients to the roots.
Vieta's Formulas – Using Vieta's formulas to find relationships between the roots and coefficients of a polynomial.
Substitution and Simplification – The process of substituting known values into equations to solve for unknowns.