Question

assessment window closed - this assessment is not currently available. please contact your test administrator. return to illuminate home connection. consider the equation $x^2 - x - 7 = 0$ a. use the discriminant to describe the solutions to the equation, including how many there are and if they are rational or irrational. b. solve the equation write your responses in the space below and use the equation editor, if needed. x=105

Explanation:

Step1: Identify quadratic coefficients

For $x^2 - x - 7 = 0$, $a=1$, $b=-1$, $c=-7$

Step2: Calculate discriminant

$\Delta = b^2 - 4ac = (-1)^2 - 4(1)(-7) = 1 + 28 = 29$

Step3: Analyze discriminant

Since $\Delta=29>0$ and 29 is not a perfect square, there are 2 distinct irrational real solutions.

Step4: Apply quadratic formula

$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{1 \pm \sqrt{29}}{2}$

Answer:

A. The equation has 2 distinct irrational real solutions, because its discriminant is 29, a positive non-perfect square number.
B. $x = \frac{1 + \sqrt{29}}{2}$ and $x = \frac{1 - \sqrt{29}}{2}$