Problem 27
Question
\(2 u^{2}+u-7=0\)
Step-by-Step Solution
Verified Answer
The solutions are \(u = \frac{-1 + \sqrt{57}}{4}\) and \(u = \frac{-1 - \sqrt{57}}{4}\).
1Step 1: Identify the quadratic equation
The given equation is already in standard quadratic form: \(2u^{2} + u - 7 = 0\)
2Step 2: Identify coefficients
In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients where: \(a = 2\), \(b = 1\), and \(c = -7\)
3Step 3: Use the quadratic formula
The quadratic formula is given by: \(u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Plug in the values of \(a\), \(b\), and \(c\): \(u = \frac{-1 \pm \sqrt{1^2 - 4(2)(-7)}}{2(2)}\) \(u = \frac{-1 \pm \sqrt{1 + 56}}{4}\)
4Step 4: Simplify under the square root
Simplify the expression inside the square root: \(u = \frac{-1 \pm \sqrt{57}}{4}\)
5Step 5: Solve for the solutions
Write the solutions in simplified form: \(u = \frac{-1 + \sqrt{57}}{4}\) and \(u = \frac{-1 - \sqrt{57}}{4}\)
Key Concepts
Roots of Equation
Roots of Equation
The roots of a quadratic equation are the values of the variable that satisfy the equation. These are also known as solutions or zeros of the equation.
By using the quadratic formula, we calculate the roots. From our original exercise:
1. We first simplify under the square root to get \[u = \frac{{-1 \pm \sqrt{57}}}{4}\]
2. Then, we solve for the two possible values to find the roots: \[u = \frac{{-1 + \sqrt{57}}}{4} \text{and} \ u = \frac{{-1 - \sqrt{57}}}{4}\]
These values represent the points where the quadratic equation intersects the x-axis.
By using the quadratic formula, we calculate the roots. From our original exercise:
1. We first simplify under the square root to get \[u = \frac{{-1 \pm \sqrt{57}}}{4}\]
2. Then, we solve for the two possible values to find the roots: \[u = \frac{{-1 + \sqrt{57}}}{4} \text{and} \ u = \frac{{-1 - \sqrt{57}}}{4}\]
These values represent the points where the quadratic equation intersects the x-axis.