Let's choose the equation \( x^2 - 16x = -28 \) to solve.
Step 1: Rewrite the equation
Rearranging the equation gives: \[ x^2 - 16x + 28 = 0 \]
Step 2: Use the Quadratic Formula
The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = -16 \), and \( c = 28 \).
Step 3: Calculate the discriminant
First, we calculate the discriminant \( b^2 - 4ac \): \[ b^2 - 4ac = (-16)^2 - 4(1)(28) = 256 - 112 = 144 \]
Step 4: Solve for x
Now we can substitute our values into the quadratic formula: \[ x = \frac{-(-16) \pm \sqrt{144}}{2(1)} = \frac{16 \pm 12}{2} \] Calculating the two potential values for \( x \):
- \( x = \frac{16 + 12}{2} = \frac{28}{2} = 14 \)
- \( x = \frac{16 - 12}{2} = \frac{4}{2} = 2 \)
Solutions
The solutions to the equation are: \[ x = 14 \quad \text{and} \quad x = 2 \]
Step 5: Find the Vertex
The vertex \( (h, k) \) of a parabola given by \( y = ax^2 + bx + c \) can be found using: \[ h = -\frac{b}{2a} = -\frac{-16}{2 \cdot 1} = \frac{16}{2} = 8 \] To find \( k \), substitute \( h = 8 \) back into the original equation: \[ k = 8^2 - 16(8) + 28 = 64 - 128 + 28 = -36 \] Thus, the vertex is: \[ (8, -36) \]
Step 6: Find the Y-intercept
To find the y-intercept, we set \( x = 0 \): \[ y = 0^2 - 16(0) + 28 = 28 \] The y-intercept is: \[ (0, 28) \]
Step 7: Summary of Results
- Vertex: \( (8, -36) \)
- X intercepts: \( (14, 0) \) and \( (2, 0) \)
- Y intercept: \( (0, 28) \)
This concludes the solution to the selected equation. If this were to be submitted via webmail, I would note "Submitted via webmail" and include the date of submission.
1 answer