To solve the equation [tex]\(-x - 3 = x^2 - 2x - 15\)[/tex] by using the graph of the system:
[tex]\[
\begin{array}{l}
y = -x - 3 \\
y = x^2 - 2x - 15
\end{array}
\][/tex]
you need to find the points where the two graphs intersect. Here’s a step-by-step process to understand and find the solutions:
1. Graph the Linear Equation:
- The equation [tex]\(y = -x - 3\)[/tex] is a linear equation.
- The slope of this line is [tex]\(-1\)[/tex] and the y-intercept is [tex]\(-3\)[/tex]. This means the line crosses the y-axis at [tex]\(-3\)[/tex] and goes downwards with a slope of [tex]\(-1\)[/tex].
2. Graph the Quadratic Equation:
- The equation [tex]\(y = x^2 - 2x - 15\)[/tex] is a quadratic equation, which forms a parabola.
- The parabola opens upwards because the coefficient of [tex]\(x^2\)[/tex] is positive.
- To better graph this, you would typically find the vertex and the x-intercepts (roots) of the parabola. However, for the intersection points, we can focus on the points where it meets the linear graph.
3. Find Intersection Points:
- To find the exact points where these graphs intersect, we set the equations equal to each other: [tex]\(-x - 3 = x^2 - 2x - 15\)[/tex].
- This results in the equation: [tex]\(x^2 - x - 12 = 0\)[/tex].
4. Solve the Quadratic Equation:
- The quadratic equation [tex]\(x^2 - x - 12 = 0\)[/tex] can be factored as:
[tex]\[
(x - 4)(x + 3) = 0
\][/tex]
- This gives the solutions:
[tex]\[
x - 4 = 0 \quad \text{or} \quad x + 3 = 0
\][/tex]
[tex]\[
x = 4 \quad \text{or} \quad x = -3
\][/tex]
5. Confirm the Points:
- Substitute these [tex]\(x\)[/tex]-values back into either of the original equations to confirm the intersection points.
- For [tex]\(x = 4\)[/tex]:
[tex]\[
y = -4 - 3 = -7
\][/tex]
- For [tex]\(x = -3\)[/tex]:
[tex]\[
y = 3 - 3 = 0
\][/tex]
Hence, the intersection points of the graphs are [tex]\((-3, 0)\)[/tex] and [tex]\((4, -7)\)[/tex]. Thus, the solutions to the equation [tex]\(-x - 3 = x^2 - 2x - 15\)[/tex] are:
[tex]\[
x = -3 \quad \text{and} \quad x = 4.
\][/tex]
Therefore, the solutions identified from the graph are:
- [tex]\(-3\)[/tex]
- [tex]\(4\)[/tex]
