Question

solve using the quadratic formula.
-2 = -4k² - 6k
write your answers as integers, proper or improper fractions in simplest form, or decimals rounded to the nearest hundredth.
k = or k =

Explanation:

Step1: Rewrite the equation in standard form

First, we need to rewrite the equation \(-2 = -4k^2 - 6k\) in the standard quadratic form \(ax^2+bx + c = 0\) (here we use \(k\) instead of \(x\)). Add \(4k^2\) and \(6k\) to both sides of the equation:
\(4k^2+6k - 2=0\)
We can simplify this equation by dividing all terms by 2:
\(2k^2 + 3k- 1=0\)
Now we have \(a = 2\), \(b=3\), \(c=- 1\) for the quadratic formula \(k=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\)

Step2: Calculate the discriminant

The discriminant \(D=b^2-4ac\). Substitute \(a = 2\), \(b = 3\), \(c=-1\) into the formula for the discriminant:
\(D=(3)^2-4\times2\times(-1)=9 + 8=17\)

Step3: Apply the quadratic formula

Now substitute \(a = 2\), \(b = 3\), and \(D = 17\) into the quadratic formula:
\(k=\frac{-3\pm\sqrt{17}}{2\times2}=\frac{-3\pm\sqrt{17}}{4}\)
We know that \(\sqrt{17}\approx4.123\)
For the plus sign:
\(k=\frac{-3 + 4.123}{4}=\frac{1.123}{4}\approx0.28\)
For the minus sign:
\(k=\frac{-3-4.123}{4}=\frac{-7.123}{4}\approx - 1.78\)

Answer:

\(k=-1.78\) or \(k = 0.28\) (or in exact form \(k=\frac{-3+\sqrt{17}}{4}\) or \(k=\frac{-3 - \sqrt{17}}{4}\))