Question

Select the quadratic equation that is equivalent to: the product of negative six and a number squared, plus the product of five and the same number is thirty-one.

Answer 1

Let's define the variable as \( x \), representing the number. According to the description, we need to express the equation based on the following components:

1. **The product of negative six and a number squared:** This can be represented as \(-6x^2\).
2. **The product of five and the same number:** This can be represented as \(5x\).
3. **Is thirty-one:** This indicates that the equation is equal to 31.

Putting these components together, we achieve the equation:

\[
-6x^2 + 5x = 31
\]

To convert this into standard quadratic form, we need to set the equation to zero. To do this, we subtract 31 from both sides:

\[
-6x^2 + 5x - 31 = 0
\]

This is the quadratic equation equivalent to the given statement. If you prefer the quadratic equation in the standard form \( ax^2 + bx + c = 0 \), we can multiply through by -1 (to make the leading coefficient positive):

\[
6x^2 - 5x + 31 = 0
\]

So the quadratic equation equivalent to the original statement is:

\[
-6x^2 + 5x - 31 = 0
\]

or

\[
6x^2 - 5x + 31 = 0
\]

You can use either form as an equivalent quadratic equation.