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Factor the expression completely. 5x^2-x^4-4 Answer Attempt 1 out of 3 square

Pergunta

Factor the expression completely.
5x^2-x^4-4
Answer Attempt 1 out of 3
square

Factor the expression completely. 5x^2-x^4-4 Answer Attempt 1 out of 3 square

Solução

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Ana ClaraMestre · Tutor por 5 anos

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To factor the expression \(5x^2 - x^4 - 4\) completely, we first rewrite it in a more standard form by arranging the terms in descending powers of \(x\):<br /><br />\[ -x^4 + 5x^2 - 4 \]<br /><br />Next, we can factor out a negative sign to make the leading coefficient positive:<br /><br />\[ -(x^4 - 5x^2 + 4) \]<br /><br />Now, we need to factor the quadratic expression inside the parentheses. We look for two numbers that multiply to \(4\) (the constant term) and add up to \(-5\) (the coefficient of the middle term). However, there are no such integer pairs that satisfy these conditions.<br /><br />Thus, we need to use the quadratic formula to find the roots of the quadratic equation \(x^4 - 5x^2 + 4 = 0\). Let \(y = x^2\), then the equation becomes:<br /><br />\[ y^2 - 5y + 4 = 0 \]<br /><br />We solve this quadratic equation using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -5\), and \(c = 4\):<br /><br />\[ y = \frac{5 \pm \sqrt{25 - 16}}{2} \]<br />\[ y = \frac{5 \pm 3}{2} \]<br /><br />This gives us two solutions for \(y\):<br /><br />\[ y = \frac{8}{2} = 4 \]<br />\[ y = \frac{2}{2} = 1 \]<br /><br />So, \(x^2 = 4\) or \(x^2 = 1\). Therefore, the solutions for \(x\) are:<br /><br />\[ x = \pm 2 \]<br />\[ x = \pm 1 \]<br /><br />Thus, the factored form of the original expression is:<br /><br />\[ -(x - 2)(x + 2)(x - 1)(x + 1) \]<br /><br />Therefore, the completely factored form of \(5x^2 - x^4 - 4\) is:<br /><br />\[ -(x - 2)(x + 2)(x - 1)(x + 1) \]
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