The provided image is a worksheet titled "Solving Quadratic Equations (A) Answers," which contains solutions to quadratic equations. Below, I will explain the general approach to solving quadratic equations and provide an overview of how these problems are solved.
General Approach to Solving Quadratic Equations
A quadratic equation is typically written in the standard form:
\[
ax^2 + bx + c = 0
\]
where \(a\), \(b\), and \(c\) are constants, and \(a \neq 0\).
There are several methods to solve quadratic equations:
1.
Factoring: If the quadratic expression can be factored into two binomials, we set each factor equal to zero and solve for \(x\).
2.
Completing the Square: This method transforms the quadratic equation into a perfect square trinomial.
3.
Quadratic Formula: The quadratic formula is:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
This formula works for all quadratic equations.
4.
Graphing: By graphing the quadratic function, the solutions are the x-intercepts of the parabola.
Explanation of Solutions in the Worksheet
The worksheet provides solutions using the
factoring method for most of the problems. Here’s a breakdown of the process for a few examples:
#### Example 1: Solve \( -x^2 + 6x + 7 = 0 \)
1. Factor the quadratic expression:
\[
-x^2 + 6x + 7 = -(x^2 - 6x - 7) = -(x - 7)(x + 1)
\]
2. Set each factor equal to zero:
\[
x - 7 = 0 \quad \text{or} \quad x + 1 = 0
\]
3. Solve for \(x\):
\[
x = 7 \quad \text{or} \quad x = -1
\]
#### Example 2: Solve \( x^2 + 13x + 36 = 0 \)
1. Factor the quadratic expression:
\[
x^2 + 13x + 36 = (x + 4)(x + 9)
\]
2. Set each factor equal to zero:
\[
x + 4 = 0 \quad \text{or} \quad x + 9 = 0
\]
3. Solve for \(x\):
\[
x = -4 \quad \text{or} \quad x = -9
\]
#### Example 3: Solve \( x^2 - x - 12 = 0 \)
1. Factor the quadratic expression:
\[
x^2 - x - 12 = (x - 4)(x + 3)
\]
2. Set each factor equal to zero:
\[
x - 4 = 0 \quad \text{or} \quad x + 3 = 0
\]
3. Solve for \(x\):
\[
x = 4 \quad \text{or} \quad x = -3
\]
#### Example 4: Solve \( x^2 - 2x + 48 = 0 \)
1. Factor the quadratic expression:
\[
x^2 - 2x + 48 = -(x + 8)(x - 6)
\]
2. Set each factor equal to zero:
\[
x + 8 = 0 \quad \text{or} \quad x - 6 = 0
\]
3. Solve for \(x\):
\[
x = -8 \quad \text{or} \quad x = 6
\]
Summary of Solutions
The worksheet provides solutions for 20 quadratic equations. Each solution involves factoring the quadratic expression and setting each factor equal to zero to find the roots. The solutions are listed next to each problem.
Final Answer
The solutions are already provided in the worksheet. If you need a specific problem explained in more detail, feel free to ask!
\boxed{\text{Solutions are provided in the worksheet.}}
Parent Tip: Review the logic above to help your child master the concept of quadratic functions worksheet.
