write in vertex form (by completing the square)y=4x2+40x+96
To write the equation in vertex form by completing the square, we need to follow these steps:
Step 1: Group the x-terms together and leave the constant term (96) separate.
y = 4x^2 + 40x + 96
Step 2: Factor out the coefficient of x^2 (4) from the x-terms.
y = 4(x^2 + 10x) + 96
Step 3: Take half of the coefficient of x (10) and square it.
Half of 10 is 5, and 5 squared is 25.
Step 4: Add the squared value obtained in Step 3 inside the parentheses, and also subtract it outside the parentheses to balance the equation.
y = 4(x^2 + 10x + 25 - 25) + 96
Step 5: Simplify the equation inside the parentheses.
y = 4((x + 5)^2 - 25) + 96
Step 6: Distribute the 4 across the parentheses.
y = 4(x + 5)^2 - 4(25) + 96
Step 7: Simplify the equation further.
y = 4(x + 5)^2 - 100 + 96
Step 8: Combine like terms.
y = 4(x + 5)^2 - 4
So, the equation is now in vertex form: y = 4(x + 5)^2 - 4, where the vertex of the parabola is given by the values h = -5 (opposite sign of the x-term) and k = -4 (constant term).