Questions: Solve the equation. Express numbers in exact form.
q^2+2q-15=0
The solution set is .
Transcript text: Solve the equation. Express numbers in exact form.
\[
q^{2}+2 q-15=0
\]
The solution set is $\square$ \}.
Solution
Solution Steps
To solve the quadratic equation \( q^2 + 2q - 15 = 0 \), we can use the quadratic formula, which is given by \( q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 1 \), \( b = 2 \), and \( c = -15 \). We will calculate the discriminant \( \Delta = b^2 - 4ac \) and then find the two possible values for \( q \).
Solution Approach
Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
Calculate the discriminant \( \Delta = b^2 - 4ac \).
Use the quadratic formula to find the roots of the equation.
Step 1: Identify the Coefficients
The given quadratic equation is \( q^2 + 2q - 15 = 0 \). From this equation, we identify the coefficients as follows:
\( a = 1 \)
\( b = 2 \)
\( c = -15 \)
Step 2: Calculate the Discriminant
We calculate the discriminant \( \Delta \) using the formula:
\[
\Delta = b^2 - 4ac
\]
Substituting the values of \( a \), \( b \), and \( c \):
\[
\Delta = 2^2 - 4 \cdot 1 \cdot (-15) = 4 + 60 = 64
\]
Step 3: Find the Roots Using the Quadratic Formula
Using the quadratic formula \( q = \frac{-b \pm \sqrt{\Delta}}{2a} \), we can find the roots:
\[
q_1 = \frac{-2 + \sqrt{64}}{2 \cdot 1} = \frac{-2 + 8}{2} = \frac{6}{2} = 3.0
\]
\[
q_2 = \frac{-2 - \sqrt{64}}{2 \cdot 1} = \frac{-2 - 8}{2} = \frac{-10}{2} = -5.0
\]
Final Answer
The solution set for the equation \( q^2 + 2q - 15 = 0 \) is:
\[
\boxed{\{3.0, -5.0\}}
\]