To work with the quadratic expression \(-16t^2 + 40t + 5\), I'll assume you want to either factor it, find its roots, or analyze it in some way. I’ll provide steps for both factoring and finding the roots using the quadratic formula.
1. Finding the Roots Using the Quadratic Formula
The quadratic formula is given by:
\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For the quadratic equation \(-16t^2 + 40t + 5\), we identify the coefficients:
- \(a = -16\)
- \(b = 40\)
- \(c = 5\)
Step 1: Calculate the Discriminant
The discriminant \(D\) is given by \(b^2 - 4ac\).
\[ D = 40^2 - 4(-16)(5) \]
Calculating this:
\[ D = 1600 + 320 = 1920 \]
Step 2: Apply the Quadratic Formula
Now plug in the values into the quadratic formula.
\[ t = \frac{-40 \pm \sqrt{1920}}{2(-16)} \]
Step 3: Simplify the Square Root
First, simplify \(\sqrt{1920}\).
Calculate the prime factorization of \(1920\):
\[ 1920 = 64 \times 30 = 64 \times (3 \times 10) = 64 \times (3 \times 2 \times 5) \]
Thus, \(1920 = 2^6 \times 3 \times 5\), and
\[ \sqrt{1920} = 8\sqrt{30} \]
Step 4: Substitute Back into the Equation
Now substituting back:
\[ t = \frac{-40 \pm 8\sqrt{30}}{-32} \]
Which simplifies to:
\[ t = \frac{40 \mp 8\sqrt{30}}{32} \]
Step 5: Factor Out Common Terms
Simplifying further:
\[ t = \frac{5 \mp \sqrt{30}}{4} \]
2. Summary of the Roots
The solutions (roots) of the equation \(-16t^2 + 40t + 5 = 0\) are:
\[ t = \frac{5 + \sqrt{30}}{4} \] \[ t = \frac{5 - \sqrt{30}}{4} \]
3. Vertex Form (if needed)
If you'd like to convert the expression to vertex form, follow through completing the square:
\[ -16(t^2 - \frac{40}{16}t) + 5 \] \[ -16(t^2 - 2.5t) + 5 \]
To complete the square:
- Take \(-2.5\), halve it to get \(-1.25\), and square it to get \(1.5625\).
- Adjust for the constant:
\[ -16(t^2 - 2.5t + 1.5625 - 1.5625) + 5 \] \[ -16((t - 1.25)^2 - 1.5625) + 5 \] \[ = -16(t - 1.25)^2 + 25 + 5 \] \[ = -16(t - 1.25)^2 + 30 \]
So the vertex form is:
\[ y = -16(t - 1.25)^2 + 30 \]
Conclusion
You can find the roots, factor the expression, or convert it into vertex form depending on your needs! If you have a specific requirement, please let me know!
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