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Math  /  Algebra

QuestionSolve equation by using the quadratic formula. List the solutions, separated by comm 4k2−k−1=0k=□\begin{array}{l} 4 k^{2}-k-1=0 \\ k=\square \end{array} Question Help: Video Message instructor Submit Question

Studdy Solution

STEP 1

1. The equation 4k2−k−1=0 4k^2 - k - 1 = 0 is a quadratic equation in standard form ax2+bx+c=0 ax^2 + bx + c = 0 .
2. The quadratic formula can be used to find the solutions of the equation, which is given by k=−b±b2−4ac2a k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .

STEP 2

1. Identify the coefficients a a , b b , and c c .
2. Calculate the discriminant b2−4ac b^2 - 4ac .
3. Apply the quadratic formula to find the solutions.
4. List the solutions.

STEP 3

Identify the coefficients a a , b b , and c c from the quadratic equation 4k2−k−1=0 4k^2 - k - 1 = 0 .
- a=4 a = 4 - b=−1 b = -1 - c=−1 c = -1

STEP 4

Calculate the discriminant b2−4ac b^2 - 4ac .
b2−4ac=(−1)2−4⋅4⋅(−1) b^2 - 4ac = (-1)^2 - 4 \cdot 4 \cdot (-1) b2−4ac=1+16 b^2 - 4ac = 1 + 16 b2−4ac=17 b^2 - 4ac = 17

STEP 5

Apply the quadratic formula k=−b±b2−4ac2a k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} .
k=−(−1)±172⋅4 k = \frac{-(-1) \pm \sqrt{17}}{2 \cdot 4} k=1±178 k = \frac{1 \pm \sqrt{17}}{8}

STEP 6

List the solutions.
The solutions are:
k=1+178,k=1−178 k = \frac{1 + \sqrt{17}}{8}, \quad k = \frac{1 - \sqrt{17}}{8}
The solutions of the quadratic equation are:
1+178,1−178 \boxed{\frac{1 + \sqrt{17}}{8}, \frac{1 - \sqrt{17}}{8}}

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