y = 4x2 - 12x + 9
You want zeros of y or what?
if zeros:
4 x^2 - 12 x + 9 = 0
(2 x-3)(2x-3) = 0
x = 1.5, 1.5 double root
If vertex etc:
4 x^2 -12 x = y - 9
x^2 - 3 y = (1/4)(y-9)
x^2 - 3 y + 9/4 = (1/4)y
(x-1.5)^2 = y/4
vertex at (1.5 , 0) which we knew already
To solve the equation y = 4x^2 - 12x + 9, you can follow these steps:
Step 1: Set y equal to zero.
0 = 4x^2 - 12x + 9
Step 2: Factor the equation if possible. However, in this case, the equation cannot be factored easily.
Step 3: Use the quadratic formula to find the values of x. The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the equation 4x^2 - 12x + 9 = 0, the values of a, b, and c are:
a = 4
b = -12
c = 9
Substituting these values into the quadratic formula, we get:
x = (12 ± √((-12)^2 - 4 * 4 * 9)) / (2 * 4)
Simplifying further:
x = (12 ± √(144 - 144)) / 8
x = (12 ± √(0)) / 8
x = (12 ± 0) / 8
So x = 12/8 = 3/2 = 1.5
Therefore, the equation y = 4x^2 - 12x + 9 has a single solution x = 1.5.
The given expression is a quadratic equation in terms of x:
y = 4x^2 - 12x + 9
To find the solutions or roots of the equation, we need to set y equal to zero and solve for x.
0 = 4x^2 - 12x + 9
Now, we can use the quadratic formula to find the solutions. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
Comparing this with our equation, we have a = 4, b = -12, and c = 9.
Plugging in the values into the quadratic formula, we get:
x = (-(-12) ± sqrt((-12)^2 - 4*4*9)) / (2*4)
Simplifying further:
x = (12 ± sqrt(144 - 144)) / 8
x = (12 ± sqrt(0)) / 8
Since the discriminant (b^2 - 4ac) is zero, the quadratic equation has a single real root, which is:
x = 12 / 8
or
x = 3 / 2
So, the solution to the quadratic equation y = 4x^2 - 12x + 9 is x = 3/2.