In the expression x² - 7x - 18, what does the coefficient -7 represent?

In the expression x² - 7x - 18, what does the coefficient -7 represent?

• A. The sum of the roots
• B. The product of the roots
• C. The leading coefficient
• D. The discriminant

Answer: (A)

In the expression \( x^2 - 7x - 18 \), the coefficient of the linear term, which is -7, actually represents the sum of the roots when the quadratic expression is set to zero. This follows from Vieta's formulas, which state that for a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots is given by \( -\frac{b}{a} \). In this case, the quadratic is in the form where \( a = 1 \) and \( b = -7 \). Thus, the sum of the roots is calculated as: \[ -\frac{-7}{1} = 7 \] This demonstrates that the sum of the roots of the equation \( x^2 - 7x - 18 = 0 \) is 7, and the coefficient -7 itself is the value that, when adjusted according to Vieta's formulas, yields this sum. The other concepts—such as the product of the roots, the leading coefficient, and the discriminant—are defined differently and do not correspond directly to the value of -7 in this expression. The leading coefficient is the coefficient of the \( x

In the expression ( x^2 - 7x - 18 ), the coefficient of the linear term, which is -7, actually represents the sum of the roots when the quadratic expression is set to zero. This follows from Vieta's formulas, which state that for a quadratic equation of the form ( ax^2 + bx + c = 0 ), the sum of the roots is given by ( -\frac{b}{a} ).

In this case, the quadratic is in the form where ( a = 1 ) and ( b = -7 ). Thus, the sum of the roots is calculated as:

[

-\frac{-7}{1} = 7

]

This demonstrates that the sum of the roots of the equation ( x^2 - 7x - 18 = 0 ) is 7, and the coefficient -7 itself is the value that, when adjusted according to Vieta's formulas, yields this sum.

The other concepts—such as the product of the roots, the leading coefficient, and the discriminant—are defined differently and do not correspond directly to the value of -7 in this expression. The leading coefficient is the coefficient of the ( x