Find the zeros of the following quadratic function by factoring. What are the x-intercepts of the graph of the function?
g(x) = x(x+1) + 24
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The given function is already in factored form. To find the zeros of the function, we set the function equal to zero and solve for x:
x(x+1) + 24 = 0
x^2 + x + 24 = 0
This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 1, and c = 24.
This equation does not factor nicely, so we can use the quadratic formula to find the zeros:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
x = [-1 ± sqrt((1)^2 - 4(1)(24))] / (2*1)
x = [-1 ± sqrt(1 - 96)] / 2
x = [-1 ± sqrt(-95)] / 2
Since the square root of a negative number is not a real number, this equation has no real solutions. Therefore, the function g(x) = x(x+1) + 24 has no zeros and the graph of the function has no x-intercepts.
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