Answer:
To find an expression that is equivalent to `y = x^2 + 8x + 18`, we can try to complete the square, factorize, or use the quadratic formula.
Completing the square:
We can add and subtract (8/2)^2 = 16 to the right-hand side of the equation to get:
y = x^2 + 8x + 18 + 16 - 16
y = (x + 4)^2 + 2
So an equivalent expression for `y = x^2 + 8x + 18` is `y = (x + 4)^2 + 2`.
Factorizing:
We can try to factorize the quadratic expression on the right-hand side of the equation. To do this, we need to find two numbers whose sum is 8 and whose product is 18. These numbers are 2 and 6, so we can write:
y = x^2 + 2x + 6x + 18
y = (x^2 + 2x) + (6x + 18)
y = x(x + 2) + 6(x + 3)
y = (x + 2)(x + 6)
So an equivalent expression for `y = x^2 + 8x + 18` is `y = (x + 2)(x + 6)`.
Using the quadratic formula:
We can use the quadratic formula to solve for x in the equation `y = x^2 + 8x + 18`. The quadratic formula states that for an equation of the form `ax^2 + bx + c = 0`, the solutions for x are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In our case, a = 1, b = 8, and c = 18. Substituting these values into the quadratic formula, we get:
x = (-8 ± sqrt(8^2 - 4(1)(18))) / 2(1)
x = (-8 ± sqrt(4)) / 2
x = -4 ± 1
So the solutions for x are x = -5 and x = -3. If we substitute these values into the original equation `y = x^2 + 8x + 18`, we get:
When x = -5, y = (-5)^2 + 8(-5) + 18 = 5 - 40 + 18 = -17
When x = -3, y = (-3)^2 + 8(-3) + 18 = 9 - 24 + 18 = 3
So an equivalent expression for `y = x^2 + 8x + 18` is `y = (x + 5)(x - 3)`.