Question

Solve −5k
2
+4=−8k using the quadratic formula.

k=−
5
6
​
, −
5
2
​


k=−
5
4
​
+
−16
​
, −
5
4
​
−
−16
​


k=−
5
4
​
+
−10
−16
​

​
, −
5
4
​
−
−10
−16
​

​


k=−
5
2
​
, 2

Answer 1

To solve the equation \(-5k^2 + 4 = -8k\) using the quadratic formula, we first rewrite the equation in standard form \(ax^2 + bx + c = 0\).

1. **Rearranging the equation**:
\[
-5k^2 + 8k + 4 = 0
\]

We can multiply the entire equation by \(-1\) to make the leading coefficient positive:
\[
5k^2 - 8k - 4 = 0
\]

2. **Identifying coefficients**:
Here, \(a = 5\), \(b = -8\), and \(c = -4\).

3. **Using the quadratic formula**:
The quadratic formula is given by:
\[
k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Plugging in the values of \(a\), \(b\), and \(c\):
- First calculate \(b^2 - 4ac\):
\[
b^2 = (-8)^2 = 64
\]
\[
4ac = 4 \times 5 \times (-4) = -80
\]
\[
b^2 - 4ac = 64 - (-80) = 64 + 80 = 144
\]

4. **Calculating the roots**:
Now substitute back into the quadratic formula:
\[
k = \frac{-(-8) \pm \sqrt{144}}{2 \cdot 5}
\]
\[
k = \frac{8 \pm 12}{10}
\]

This gives us two solutions:
- For \(k = \frac{8 + 12}{10}\):
\[
k = \frac{20}{10} = 2
\]
- For \(k = \frac{8 - 12}{10}\):
\[
k = \frac{-4}{10} = -\frac{2}{5}
\]

5. **Final solutions**:
Therefore, the solutions are:
\[
k = 2 \quad \text{and} \quad k = -\frac{2}{5}
\]

The solution presented in your message appears to be incorrect. The true solutions are \(\mathbf{k = 2}\) and \(\mathbf{k = -\frac{2}{5}}\).