Final answer:
To rewrite the quadratic expression y = -16x² + 32t + 10 in vertex form, one must complete the square. This involves factoring out the coefficient of the x² term, completing the square for the x term, and then properly combining all terms to reflect the original equation.
Step-by-step explanation:
To rewrite the expression y = -16x² + 32t + 10 in vertex form, we use the method of completing the square. The general vertex form for a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.
Let's start by isolating the quadratic and linear terms:
y = -16x² + 32t + 10
y + 16x² = 32t + 10
Next, we factor out the coefficient of the x² term:
y + 16x² = 32t + 10
y = -16(x² - 2t)
Now we need to complete the square within the parentheses by adding and subtracting the square of half the coefficient of x:
y = -16[(x - t)² - t²]
y = -16(x - t)² + 16t²
Finally, we substitute back the remainder of the initial expression:
y = -16(x - t)² + 16t² + 10
Since 16t² is a part of the original expression, we need to ensure that we substitute it properly to keep the equation equivalent to the original.