To solve the provided quadratic equations using the completing the square method, we will follow the standard steps for each equation.
1. \( x^2 - 4x = 0 \)
Step 1: Move to one side of the equation: \[ x^2 - 4x = 0 \]
Step 2: Complete the square: \[ x^2 - 4x + 4 = 4 \] Now we can rewrite it as: \[ (x - 2)^2 = 4 \]
Step 3: Solve for \(x\): \[ x - 2 = \pm 2 \] Thus, \[ x - 2 = 2 \implies x = 4 \quad \text{and} \quad x - 2 = -2 \implies x = 0 \] Final solutions: \[ x = 4 \quad \text{and} \quad x = 0 \]
2. \( x^2 + 8x - 5 = 0 \)
Step 1: Move the constant to the other side: \[ x^2 + 8x = 5 \]
Step 2: Complete the square: Take half of 8, square it: \(\left(\frac{8}{2}\right)^2 = 16\). Add 16 to both sides: \[ x^2 + 8x + 16 = 5 + 16 \] This simplifies to: \[ (x + 4)^2 = 21 \]
Step 3: Solve for \(x\): \[ x + 4 = \pm \sqrt{21} \] Thus: \[ x = -4 + \sqrt{21} \quad \text{and} \quad x = -4 - \sqrt{21} \] Final solutions: \[ x = -4 + \sqrt{21} \quad \text{and} \quad x = -4 - \sqrt{21} \]
3. \( 4x^2 + 32x + 68 = 0 \)
Step 1: Divide the whole equation by 4 to simplify: \[ x^2 + 8x + 17 = 0 \]
Step 2: Move the constant to the other side: \[ x^2 + 8x = -17 \]
Step 3: Complete the square: Take half of 8, square it: \(\left(\frac{8}{2}\right)^2 = 16\). Add 16 to both sides: \[ x^2 + 8x + 16 = -17 + 16 \] This simplifies to: \[ (x + 4)^2 = -1 \]
Step 4: Solve for \(x\): \[ x + 4 = \pm i \] Thus: \[ x = -4 + i \quad \text{and} \quad x = -4 - i \] Final solutions: \[ x = -4 + i \quad \text{and} \quad x = -4 - i \]
4. \( y = x^2 - 8x + 18 \)
To write this in vertex form \(y = a(x - h)^2 + k\):
Step 1: Complete the square for the quadratic expression: Take half of -8, square it: \(\left(\frac{-8}{2}\right)^2 = 16\). Add and subtract 16: \[ y = x^2 - 8x + 16 - 16 + 18 \] This simplifies to: \[ y = (x - 4)^2 + 2 \]
Now, the vertex form is: \[ y = (x - 4)^2 + 2 \] Final answer: \[ y = (x - 4)^2 + 2 \] This indicates the vertex of the parabola is at \((4, 2)\).
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