Convert the simplified parabolic equation y=3x^2-30x-7 to standard form.
3x^2-30x-7
3x^2 -30 x -7 +75 -75
3(x^2 -10x + 25) -82
3(x-5)^2 -82
or
y = 3(x-5)^2 -82
check that.
To convert the simplified parabolic equation y = 3x^2 - 30x - 7 to standard form, you need to complete the square.
Step 1: Move the constant term (-7) to the other side of the equation by adding 7 to both sides:
y + 7 = 3x^2 - 30x
Step 2: Factor out the leading coefficient (3) from the quadratic terms on the right side:
y + 7 = 3(x^2 - 10x)
Step 3: To complete the square, take half of the coefficient of the x-term (-10/2 = -5) and square it (-5^2 = 25). Add this value inside the parentheses and compensate by subtracting the same value multiplied by 3 (to maintain symmetry):
y + 7 + 3(25) = 3(x^2 - 10x + 25)
Simplifying further:
y + 7 + 75 = 3(x^2 - 10x + 25)
Step 4: Write the squared binomial as a perfect square trinomial:
y + 82 = 3(x - 5)^2
Now, the equation is in standard form: y + 82 = 3(x - 5)^2