Problem 40 - Elementary Algebra

Question

\(4 v^{2}+5 v-12=0\)

Step-by-Step Solution

Answer

\(v = \frac{-5 \pm \sqrt{217}}{8}\)

Step 1: Identify the coefficients

The given equation is in the form of a quadratic equation: \(4v^2 + 5v - 12 = 0\). Identify the coefficients: \(a = 4\), \(b = 5\), and \(c = -12\).

Step 2: Calculate the discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \(D = b^2 - 4ac\). Substitute the values of \(a, b,\) and \(c\): \(D = 5^2 - 4(4)(-12) = 25 + 192 = 217\).

Step 3: Apply the quadratic formula

The solutions for the quadratic equation can be found using the quadratic formula: \(v = \frac{-b \pm \sqrt{D}}{2a}\). Since \(D = 217\), \(a = 4\), and \(b = 5\), substitute these values into the formula: \(v = \frac{-5 \pm \sqrt{217}}{8}\).

Step 4: Simplify the solutions

Write the two solutions explicitly: \(v_1 = \frac{-5 + \sqrt{217}}{8}\) and \(v_2 = \frac{-5 - \sqrt{217}}{8}\).

Key Concepts

Coefficients

In a quadratic equation, the coefficients are the numerical values that multiply the variables. A standard quadratic equation is written in the form: \(ax^2 + bx + c = 0\). Here,

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

For the equation in the exercise, \(4v^2 + 5v - 12 = 0\), we identify the coefficients as follows:

• \(a = 4\)
• \(b = 5\)
• \(c = -12\)

Understanding the coefficients is crucial because they are used in further steps to find the solutions.

Discriminant

The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) helps us determine the nature of the roots without actually solving the equation. It's calculated using the formula: \(D = b^2 - 4ac\)For our given equation with coefficients \(a = 4\), \(b = 5\), and \(c = -12\), the discriminant is calculated as: \[D = 5^2 - 4(4)(-12) = 25 + 192 = 217\]The value of the discriminant can tell us:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is one repeated real root.
• If \(D < 0\), there are no real roots, but two complex roots.

Here, \(D = 217 > 0\), so our equation has two distinct real roots.

Quadratic Formula

The quadratic formula is a way to find the solutions of a quadratic equation \(ax^2 + bx + c = 0\). It is derived from completing the square of the quadratic equation. The formula is: \[x = \frac{-b \pm \sqrt{D}}{2a}\]Where:

• \(D\) is the discriminant
• \(a\), \(b\), and \(c\) are the coefficients

In our exercise, the quadratic formula helps us find the roots of \(4v^2 + 5v - 12 = 0\). Plugging in the values \(a = 4\), \(b = 5\), and \(D = 217\), we get: \[v = \frac{-5 \pm \sqrt{217}}{8}\]This gives us the solutions in the next step.

Solving Quadratic Equations

Solving a quadratic equation means finding the values of the variable that make the equation true. Using the quadratic formula, we have: \[v = \frac{-5 \pm \sqrt{217}}{8}\]This can be split into two separate solutions:

• \(v_1 = \frac{-5 + \sqrt{217}}{8}\)
• \(v_2 = \frac{-5 - \sqrt{217}}{8}\)

These two values are the solutions of the quadratic equation \(4v^2 + 5v - 12 = 0\). Make sure to substitute these back into the original equation to verify their correctness.