To solve the equation \( x^2 + 2x = 3 \) using a function graph, we'll first rewrite the equation in standard form:
\[ x^2 + 2x - 3 = 0 \]
Next, we can express this as a function:
\[ f(x) = x^2 + 2x - 3 \]
We need to find the roots of this function, which are the values of \( x \) where the graph of \( f(x) \) intersects the x-axis (where \( f(x) = 0 \)).
Steps to Create the Graph
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Identify the Function: \( f(x) = x^2 + 2x - 3 \)
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Find the Roots: To find the points where the graph intersects the x-axis, we can either factor the quadratic or use the quadratic formula.
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Find the roots using factoring: \[ f(x) = (x + 3)(x - 1) \] Thus, the roots are: \[ x + 3 = 0 \Rightarrow x = -3 \] \[ x - 1 = 0 \Rightarrow x = 1 \]
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Alternatively, using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1, b = 2, c = -3 \): \[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-3)}}{2 \cdot 1} \] \[ x = \frac{-2 \pm \sqrt{4 + 12}}{2} \] \[ x = \frac{-2 \pm \sqrt{16}}{2} \] \[ x = \frac{-2 \pm 4}{2} \] \[ x = 1 \quad \text{or} \quad x = -3 \]
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Graph the Function: To graph the function \( f(x) = x^2 + 2x - 3 \), we can plot key points. The vertex, which gives us the minimum point of the quadratic, can be found using:
\[ x = -\frac{b}{2a} = -\frac{2}{2} = -1 \] Substitute \( x = -1 \) back into the function to find the corresponding y-value: \[ f(-1) = (-1)^2 + 2(-1) - 3 = 1 - 2 - 3 = -4 \]
This means the vertex is at \((-1, -4)\).
Example Plot
- You can plot a few more points around this vertex, such as:
- \( f(-3) = 0 \)
- \( f(0) = -3 \)
- \( f(1) = 0 \)
Conclusion
The solutions to the equation \( x^2 + 2x = 3 \) are the x-coordinates where the function \( f(x) = x^2 + 2x - 3 \) intersects the x-axis, which are:
- \( x = -3 \)
- \( x = 1 \)
You can use graphing software or a graphing calculator to visually represent and confirm these intersections.