Question

question 1
which two values of x are roots of the polynomial
below?
x² - 11x + 15
a. x = 3
b. x = (11 - √(-109))/4
c. x = (11 + √61)/2
d. x = 2.5
e. x = (11 - √61)/2
f. x = (11 + √(-109))/4

question 2
which two values of x are roots of the polynomial
below?
x² + 5x + 7
a. x = (5 - √17)/2
b. x = (5 + √17)/2
c. x = 5
d. x = 1/2
e. x = (-5 - √(-3))/2
f. x = (-5 + √(-3))/2

question 3
which two values of x are roots of the polynomial
below?
x² + 5x + 9
a. x = (-5 + √61)/2
b. x = (5 - √17)/2
c. x = (5 + √17)/2
d. x = (-5 - √61)/2
e. x = (-5 - √(-11))/2
f. x = (-5 + √(-11))/2

question 4
which two values of x are roots of the polynomial
below?
x² + 5x + 11
a. x = (-5 + √69)/2
b. x = (5 - √17)/2
c. x = (5 + √17)/2
d. x = (-5 - √69)/2
e. x = (-5 - √(-19))/2
f. x = (-5 + √(-19))/2

Explanation:

Question 1

To find the roots of the quadratic polynomial \(x^2 - 11x + 15\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) for a quadratic equation \(ax^2+bx + c = 0\). Here, \(a = 1\), \(b=- 11\), and \(c = 15\).

Step 1: Calculate the discriminant \(\Delta=b^2-4ac\)

\(\Delta=(-11)^2-4\times1\times15=121 - 60 = 61\)

Step 2: Apply the quadratic formula

\(x=\frac{-(-11)\pm\sqrt{61}}{2\times1}=\frac{11\pm\sqrt{61}}{2}\)
So the roots are \(x=\frac{11 + \sqrt{61}}{2}\) (Option C) and \(x=\frac{11-\sqrt{61}}{2}\) (Option E)

For the quadratic polynomial \(x^2+5x + 7\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = 5\), and \(c=7\).

Step 1: Calculate the discriminant \(\Delta=b^2-4ac\)

\(\Delta=5^2-4\times1\times7=25 - 28=- 3\)

Step 2: Apply the quadratic formula

Since \(\Delta=-3\), we have \(x=\frac{-5\pm\sqrt{-3}}{2}\)
So the roots are \(x=\frac{-5-\sqrt{-3}}{2}\) (Option E) and \(x=\frac{-5+\sqrt{-3}}{2}\) (Option F)

For the quadratic polynomial \(x^2 + 5x+9\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) with \(a = 1\), \(b = 5\), and \(c = 9\).

Step 1: Calculate the discriminant \(\Delta=b^2-4ac\)

\(\Delta=5^2-4\times1\times9=25 - 36=-11\)

Step 2: Apply the quadratic formula

\(x=\frac{-5\pm\sqrt{-11}}{2}\)
So the roots are \(x=\frac{-5-\sqrt{-11}}{2}\) (Option E) and \(x=\frac{-5+\sqrt{-11}}{2}\) (Option F)

Answer:

C. \(x=\frac{11+\sqrt{61}}{2}\), E. \(x=\frac{11 - \sqrt{61}}{2}\)

Question 2