Questions: Solve the equation. Express numbers in exact form. q^2+2q-15=0 The solution set is .

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 $\square$ \}.
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Solution

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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
  1. Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation.
  2. Calculate the discriminant \( \Delta = b^2 - 4ac \).
  3. 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: \[ \boxed{\{3.0, -5.0\}} \]

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