Colon Lindsey
07/29/2023 · Primary School
Writing a quadratic equation given the roots and the leading coefficient (Use the letter \( x \) to represent the variable.) \( \square=0 \)
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Step-by-step Solution
To write a quadratic equation given the roots and the leading coefficient, you can use the fact that if \( r_1 \) and \( r_2 \) are the roots of the quadratic equation, then the equation can be expressed in the form:
\[
a(x - r_1)(x - r_2) = 0
\]
where \( a \) is the leading coefficient.
Let's say the roots are \( r_1 \) and \( r_2 \), and the leading coefficient is \( a \). The quadratic equation can be expanded as follows:
1. Start with the factored form:
\[
a(x - r_1)(x - r_2) = 0
\]
2. Expand the factors:
\[
a[(x - r_1)(x - r_2)] = a[x^2 - (r_1 + r_2)x + r_1r_2]
\]
3. Distribute the leading coefficient \( a \):
\[
ax^2 - a(r_1 + r_2)x + ar_1r_2 = 0
\]
So, the final form of the quadratic equation is:
\[
ax^2 - a(r_1 + r_2)x + ar_1r_2 = 0
\]
If you provide specific values for the roots \( r_1 \), \( r_2 \), and the leading coefficient \( a \), I can help you write the specific quadratic equation.
Quick Answer
The quadratic equation with roots \( r_1 \) and \( r_2 \) and leading coefficient \( a \) is \( ax^2 - a(r_1 + r_2)x + ar_1r_2 = 0 \).
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