Solve for f.

21f^2–28f–105=0
Write each solution as an integer, proper fraction, or improper fraction in simplest form. If there are multiple solutions, separate them with commas.
f=

1 answer

To solve the quadratic equation \(21f^2 - 28f - 105 = 0\), we can use the quadratic formula:

\[ f = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

where \(a = 21\), \(b = -28\), and \(c = -105\).

First, we calculate the discriminant (\(b^2 - 4ac\)):

\[ b^2 = (-28)^2 = 784 \] \[ 4ac = 4 \cdot 21 \cdot (-105) = -8820 \] \[ b^2 - 4ac = 784 - (-8820) = 784 + 8820 = 9604 \]

Next, we find the square root of the discriminant:

\[ \sqrt{9604} = 98 \]

Now, we use the quadratic formula:

\[ f = \frac{-(-28) \pm 98}{2 \cdot 21} \] \[ f = \frac{28 \pm 98}{42} \]

This gives us two cases:

  1. First Case: \(f = \frac{28 + 98}{42} = \frac{126}{42} = 3\)
  2. Second Case: \(f = \frac{28 - 98}{42} = \frac{-70}{42} = \frac{-5}{3}\)

Thus, the solutions for \(f\) are:

\[ \boxed{3, -\frac{5}{3}} \]

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