Answer:
8
Explanation:
Find the x-intercept of f(x) = x² - 16x + 64
The x-intercept of the function occurs when y=0, which is when the graph touches the x-axis. To find it, factor the equation.
For this function, you do not need to use the quadratic formula.
Note the trinomial in standard form f(x) = ax² + bx + c
f(x) = x² - 16x + 64 looks like a perfect trinomial. Perfect trinomials are a special trinomial that follow a pattern you can use to factor.
You know a trinomial is perfect when two times the square root of the first term and the square root of the last to is equal to the absolute value of the middle term. (It does not matter if "b" is negative or positive).
As an equation:
|bx| = 2 √(ax²) √c
In this equation:
ax² = x²
|bx| = 16x
c = 64
Substitute the values into our equation. Check if left side equals right side (LS = RS).
|bx| = 2 √(ax²) √c
16x = 2 √(x²) √64
16x = 2*x*8
16x = 16x
LS = RS
The equation is a perfect trinomial.
To factor perfect trinomials, use the rule:
ax² ± bx + c = (√(ax²) ± √c)²
Whether the ± in factored form is plus or minus depends on if "b" is positive or negative.
Substitute the function and find the factored form.
f(x) = ax² ± bx + c = (√(ax²) ± √c)²
x² - 16x + 64 = (√(x²) - √64)² Eliminate reverse operations and simplify
= (x - 8)² Factored form
To find the x - intercept, make y = 0, because that's when the x-intercept occurs. Isolate "x" in reverse BEDMAS order with reverse operations. (Reverse operations "cancel out" a number because the opposite is done.)
f(x) = (x - 8)² "f(x)" is the same as "y"
y = (x - 8)² Substitute "y" for 0
0 = (x - 8)²
√0 = √(x - 8)² Square root both sides
0 = x - 8
0 + 8 = x - 8 + 8 Add 8 to both sides to cancel out the -8 on the left side
8 = x
x = 8 Writing the variable on the left side is standard formatting
Therefore the x-intercept of f(x) = x² - 16x + 64 is 8.