To find the roots and vertex of the quadratic equation \( y = x^2 + 2x - 120 \), we can use the quadratic formula and vertex formula.
Step 1: Finding the Roots
The quadratic formula for finding the roots of \( ax^2 + bx + c = 0 \) is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the equation \( y = x^2 + 2x - 120 \):
- \( a = 1 \)
- \( b = 2 \)
- \( c = -120 \)
Now, we calculate the discriminant: \[ b^2 - 4ac = 2^2 - 4(1)(-120) = 4 + 480 = 484 \]
Now apply the quadratic formula: \[ x = \frac{-2 \pm \sqrt{484}}{2(1)} = \frac{-2 \pm 22}{2} \]
Calculating the two potential roots:
- \( x_1 = \frac{-2 + 22}{2} = \frac{20}{2} = 10 \)
- \( x_2 = \frac{-2 - 22}{2} = \frac{-24}{2} = -12 \)
So the roots of the quadratic are: \[ x_1 = 10 \quad \text{and} \quad x_2 = -12 \]
Step 2: Finding the Vertex
The vertex of a quadratic \( y = ax^2 + bx + c \) can be found using the formula: \[ x = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \): \[ x = -\frac{2}{2(1)} = -\frac{2}{2} = -1 \]
Now, to find the y-coordinate of the vertex, substitute \( x = -1 \) back into the original function: \[ y = (-1)^2 + 2(-1) - 120 = 1 - 2 - 120 = -121 \]
Final Results
Thus, the roots of the quadratic are:
- \( x_1 = 10 \)
- \( x_2 = -12 \)
The vertex of the quadratic is:
- Vertex: \( (-1, -121) \)
Summary
- Roots: \( 10 \) and \( -12 \)
- Vertex: \( (-1, -121) \)
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