7x^2-12x-4

To analyze the quadratic expression \(7x^2 - 12x - 4\), we can perform several operations, such as factoring (if possible), finding the roots using the quadratic formula, or determining properties like the vertex and axis of symmetry.

Step 1: Find the Roots Using the Quadratic Formula

The quadratic formula is given by:

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

For the given quadratic \(7x^2 - 12x - 4\), we have:

• \(a = 7\)
• \(b = -12\)
• \(c = -4\)

Now, calculate the discriminant:

\[ b^2 - 4ac = (-12)^2 - 4(7)(-4) = 144 + 112 = 256 \]

Now substitute back into the formula:

\[ x = \frac{-(-12) \pm \sqrt{256}}{2(7)} = \frac{12 \pm 16}{14} \]

This gives us two potential solutions:

1. \(x = \frac{12 + 16}{14} = \frac{28}{14} = 2\)
2. \(x = \frac{12 - 16}{14} = \frac{-4}{14} = -\frac{2}{7}\)

Step 2: Factoring the Quadratic

To factor \(7x^2 - 12x - 4\), we look for two numbers that multiply to \(a \cdot c = 7 \cdot -4 = -28\) and add to \(b = -12\).

After checking, we can use the roots found earlier to express the quadratic in factored form:

\[ 7x^2 - 12x - 4 = 7(x - 2)(x + \frac{2}{7}) \]

Conclusion

The quadratic \(7x^2 - 12x - 4\) has roots \(x = 2\) and \(x = -\frac{2}{7}\), and can be factored (although it is not in simplest integer form) as \(7(x - 2)(x + \frac{2}{7})\).

If you need more specific information or further analysis on the quadratic, feel free to ask!