𝑥 2 + 4𝑥 − 21 = 0 𝑥 2 + 8𝑥 − 20 = 0 What are the values of a=________ b= ________ c= ________ What are the values of a=________ b= ________ c= ________ Using any method, what are the roots? x1= _________ x2= _________ Using any method, what are the roots? x1= _________ x2= _________ Perform the indicated operations x1 + x2 = _________ (x1) • (x2)= _________ Perform the indicated operations x1 + x2 = _________ (x1) • (x2)= _________

We have two quadratic equations to analyze and solve for the coefficients \(a\), \(b\), and \(c\), as well as their respective roots.

### First equation:
\[
x^2 + 4x - 21 = 0
\]
In this quadratic equation, we identify the coefficients:
- \( a = 1 \)
- \( b = 4 \)
- \( c = -21 \)

Using the **quadratic formula**:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
For the first equation:
\[
x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-21)}}{2(1)} = \frac{-4 \pm \sqrt{16 + 84}}{2} = \frac{-4 \pm \sqrt{100}}{2}
\]
\[
x = \frac{-4 \pm 10}{2}
\]
Thus, the roots are:
- \( x_1 = \frac{-4 + 10}{2} = 3 \)
- \( x_2 = \frac{-4 - 10}{2} = -7 \)

Now, performing the operations:
- \( x_1 + x_2 = 3 + (-7) = -4 \)
- \( x_1 \cdot x_2 = 3 \cdot (-7) = -21 \)

### Second equation:
\[
x^2 + 8x - 20 = 0
\]
In this quadratic equation, we identify the coefficients:
- \( a = 1 \)
- \( b = 8 \)
- \( c = -20 \)

Using the quadratic formula:
\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-20)}}{2(1)} = \frac{-8 \pm \sqrt{64 + 80}}{2} = \frac{-8 \pm \sqrt{144}}{2}
\]
\[
x = \frac{-8 \pm 12}{2}
\]
Thus, the roots are:
- \( x_1 = \frac{-8 + 12}{2} = 2 \)
- \( x_2 = \frac{-8 - 12}{2} = -10 \)

Now, performing the operations:
- \( x_1 + x_2 = 2 + (-10) = -8 \)
- \( x_1 \cdot x_2 = 2 \cdot (-10) = -20 \)

### Summary:
#### First Equation:
- \( a = 1 \), \( b = 4 \), \( c = -21 \)
- Roots: \( x_1 = 3 \), \( x_2 = -7 \)
- Operations: 
  - \( x_1 + x_2 = -4 \)
  - \( x_1 \cdot x_2 = -21 \)

#### Second Equation:
- \( a = 1 \), \( b = 8 \), \( c = -20 \)
- Roots: \( x_1 = 2 \), \( x_2 = -10 \)
- Operations:
  - \( x_1 + x_2 = -8 \)
  - \( x_1 \cdot x_2 = -20 \)

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Would you like further clarification on any of these steps?

Here are 5 related questions to further explore this topic:
1. How can we use factoring to solve quadratic equations like these?
2. What is the geometric significance of the sum and product of the roots?
3. How can you derive the quadratic formula from completing the square?
4. How do discriminants help determine the nature of the roots in quadratic equations?
5. How do different methods (factoring, completing the square) compare to the quadratic formula?

**Tip:** The discriminant (\(b^2 - 4ac\)) tells you whether the roots of a quadratic equation are real and distinct, real and equal, or complex.

𝑥^2 + 4𝑥 − 21 = 0, 𝑥^2 + 8𝑥 − 20 = 0. What are the values of a=________, b= ________, c= ________? What are the roots? x1= _________, x2= _________. Perform the indicated operations x1 + x2 = _________, (x1) • (x2)= ________.

This page offers a step-by-step solution for solving two quadratic equations: x^2 + 4x - 21 = 0 and x^2 + 8x - 20 = 0. Learn how to apply the quadratic formula to find the roots and perform operations on the solutions. Suitable for students in Grades 9-10.

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