Complete the square to re-write the quadratic function in vertex form:

Question

y, equals, minus, 7, x, squared, minus, 70, x, minus, 172
y=−7x
2
−70x−172

Answer 1

To rewrite the quadratic function \( y = -7x^2 - 70x - 172 \) in vertex form by completing the square, follow these steps:

1. Factor out the coefficient of \( x^2 \) from the \( x^2 \) and \( x \) terms:

   \[ y = -7(x^2 + 10x) - 172 \]

2. Complete the square inside the parentheses:

   • Take the coefficient of \( x \) (which is 10), divide it by 2, and square it: \[ \left(\frac{10}{2}\right)^2 = 5^2 = 25 \]

   • Now add and subtract this square inside the parentheses: \[ y = -7(x^2 + 10x + 25 - 25) - 172 \]

   • This simplifies to: \[ y = -7((x + 5)^2 - 25) - 172 \]

3. Distribute the \(-7\) and simplify: \[ y = -7(x + 5)^2 + 175 - 172 \] \[ y = -7(x + 5)^2 + 3 \]

4. Write the final vertex form: \[ y = -7(x + 5)^2 + 3 \]

So, the vertex form of the quadratic function is \( y = -7(x + 5)^2 + 3 \).