To factor the expression \( 5f^2 - 24f - 5 \) completely, we can follow these steps.
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Identify the coefficients: We have \( a = 5 \), \( b = -24 \), and \( c = -5 \).
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Use the quadratic formula to find roots if simple factoring doesn't seem easy: \[ f = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \] Plugging in the values: \[ f = \frac{{24 \pm \sqrt{{(-24)^2 - 4 \cdot 5 \cdot (-5)}}}}{2 \cdot 5} \] \[ = \frac{{24 \pm \sqrt{{576 + 100}}}}{10} \] \[ = \frac{{24 \pm \sqrt{676}}}{10} \] \[ = \frac{{24 \pm 26}}{10} \]
This gives us: \[ f_1 = \frac{{50}}{10} = 5 \] \[ f_2 = \frac{{-2}}{10} = -0.2 \]
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Express the quadratic as: \[ 5f^2 - 24f - 5 = 5(f - 5)(f + 0.2) \]
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Rewrite the fractional root as an integer by multiplying through by 10: \[ 5(f - 5)(10f + 2) \]
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Final factored form: The expression can be factored completely as: \[ 5(f - 5)(10f + 2) \]
To summarize, the completely factored form of \( 5f^2 - 24f - 5 \) is: \[ 5(f - 5)(10f + 2) \]