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Question: x²-4x+24 factor

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To factor the quadratic expression $$x^2 - 4x + 24$$, we start by finding its roots using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ For the given quadratic $$x^2 - 4x + 24$$: - $$a = 1$$ - $$b = -4$$ - $$c = 24$$ Plugging these values into the quadratic formula: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1}$ $x = \frac{4 \pm \sqrt{16 - 96}}{2}$ $x = \frac{4 \pm \sqrt{-80}}{2}$ $x = \frac{4 \pm \sqrt{-(16 \cdot 5)}}{2}$ $x = \frac{4 \pm 4i\sqrt{5}}{2}$ $x = 2 \pm 2i\sqrt{5}$ Since the roots $$2 + 2i\sqrt{5}$$ and $$2 - 2i\sqrt{5}$$ involve imaginary numbers, the quadratic expression does not factor easily over the reals. However, we can express it in terms of its factors involving complex numbers: $x^2 - 4x + 24 = (x - (2 + 2i\sqrt{5}))(x - (2 - 2i\sqrt{5}))$ While this is the factorization involving complex roots, it cannot be factored further into simpler polynomial expressions with real coefficients. So, over the real numbers, the quadratic expression $$x^2 - 4x + 24$$ does not factor any further.

May 24, 2024, 6:01 a.m.

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