Questions: 3. Write the quadratic equation whose roots are 2 and 3 , and whose leading coefficient is 4 .

3. Write the quadratic equation whose roots are 2 and 3 , and whose leading coefficient is 4 .
Transcript text: 3. Write the quadratic equation whose roots are 2 and 3 , and whose leading coefficient is 4 .
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Solution

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Solution Steps

Step 1: Identify the given parameters

The given roots are \(r_1 = 2\) and \(r_2 = 3\), and the leading coefficient is \(a = 4\).

Step 2: Apply the formula to construct the quadratic equation

Given the roots \(r_1\) and \(r_2\), and the leading coefficient \(a\), the quadratic equation can be written as: \[a(x - r_1)(x - r_2) = 0\] Expanding this, we get: \[a(x^2 - (r_1 + r_2)x + r_1r_2) = 0\] Or equivalently: \[ax^2 - a(r_1 + r_2)x + ar_1r_2 = 0\]

Step 3: Substitute the given values into the formula

Substituting \(r_1 = 2\), \(r_2 = 3\), and \(a = 4\) into the formula, we get: \[4(x^2 - (2 + 3)x + 2*3) = 0\] Which simplifies to: \[4x^2 -20x +24 = 0\]

Final Answer:

The quadratic equation constructed from the given roots \(r_1 = 2\) and \(r_2 = 3\), and leading coefficient \(a = 4\) is: \[4x^2 - 20x + 24 = 0\]

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