Find the zeros of the following quadratic function by completing the square. What are the x-intercepts of the graph of the function?
g(x) = x(x + 11) + 24
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First, we need to rewrite the function in standard form:
g(x) = x^2 + 11x + 24
To find the zeros of the function, we need to solve the equation x^2 + 11x + 24 = 0.
We can complete the square to solve this equation. The first step is to move the constant term to the other side of the equation:
x^2 + 11x = -24
Next, we take half of the coefficient of x, square it, and add it to both sides of the equation. Half of 11 is 5.5, and 5.5^2 = 30.25.
x^2 + 11x + 30.25 = -24 + 30.25
This simplifies to:
x^2 + 11x + 30.25 = 6.25
The left side of the equation is now a perfect square trinomial, which can be factored as:
(x + 5.5)^2 = 6.25
Taking the square root of both sides gives:
x + 5.5 = ± √6.25
Solving for x gives:
x = -5.5 ± √6.25
This simplifies to:
x = -5.5 ± 2.5
So the zeros of the function are x = -5.5 - 2.5 = -8 and x = -5.5 + 2.5 = -3.
Therefore, the x-intercepts of the graph of the function are (-8, 0) and (-3, 0).
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