Question
How do I find the solution to a quadratic equation with roots -11 and 5?
Answers
GPT-4o mini
To find the quadratic equation given its roots, you can use the fact that if \( r_1 \) and \( r_2 \) are the roots, the quadratic equation can be expressed in the form:
\[
y = a(x - r_1)(x - r_2)
\]
In this case, the roots are \( r_1 = -11 \) and \( r_2 = 5 \). Thus, we can substitute these values into the equation:
\[
y = a(x - (-11))(x - 5)
\]
\[
y = a(x + 11)(x - 5)
\]
Next, we can expand this expression:
1. First, multiply the binomials:
\[
(x + 11)(x - 5) = x^2 - 5x + 11x - 55 = x^2 + 6x - 55
\]
2. Therefore, the quadratic equation can be written as:
\[
y = a(x^2 + 6x - 55)
\]
If you want the simplest form, you can set \( a = 1 \), which gives us the quadratic equation:
\[
y = x^2 + 6x - 55
\]
Thus, the quadratic equation with roots -11 and 5 is:
\[
x^2 + 6x - 55 = 0
\]
\[
y = a(x - r_1)(x - r_2)
\]
In this case, the roots are \( r_1 = -11 \) and \( r_2 = 5 \). Thus, we can substitute these values into the equation:
\[
y = a(x - (-11))(x - 5)
\]
\[
y = a(x + 11)(x - 5)
\]
Next, we can expand this expression:
1. First, multiply the binomials:
\[
(x + 11)(x - 5) = x^2 - 5x + 11x - 55 = x^2 + 6x - 55
\]
2. Therefore, the quadratic equation can be written as:
\[
y = a(x^2 + 6x - 55)
\]
If you want the simplest form, you can set \( a = 1 \), which gives us the quadratic equation:
\[
y = x^2 + 6x - 55
\]
Thus, the quadratic equation with roots -11 and 5 is:
\[
x^2 + 6x - 55 = 0
\]