Problem 99 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(4 m^{2}+m-3=0\)

Step-by-Step Solution

Answer

The solutions are \(m = \frac{3}{4}\) and \(m = -1\).

Step 1: Identify coefficients

In the quadratic equation, identify the coefficients: \(a\), \(b\), and \(c\) from the standard form \(ax^2 + bx + c = 0\). For the equation \(4m^2 + m - 3 = 0\), \(a = 4\), \(b = 1\), and \(c = -3\).

Step 2: Write the Quadratic Formula

Recall the Quadratic Formula: \(m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Substitute the coefficients into the formula

Substitute \(a = 4\), \(b = 1\), and \(c = -3\) into the Quadratic Formula: \(m = \frac{-(1) \pm \sqrt{(1)^2 - 4(4)(-3)}}{2(4)}\).

Step 4: Simplify under the square root

Calculate the discriminant: \(1^2 - 4 \cdot 4 \cdot -3 = 1 + 48 = 49\). So, the equation becomes \(m = \frac{-1 \pm \sqrt{49}}{8}\).

Step 5: Calculate the square root

Simplify the square root: \(\sqrt{49} = 7\). The equation is now \(m = \frac{-1 \pm 7}{8}\).

Step 6: Split into two solutions

Solve for the two possible values: 1) \(m = \frac{-1 + 7}{8} = \frac{6}{8} = \frac{3}{4}\) 2) \(m = \frac{-1 - 7}{8} = \frac{-8}{8} = -1\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two. They are generally written in the form:

\[ax^2 + bx + c = 0\].
In the given exercise, we are working with the quadratic equation, \[4m^2 + m - 3 = 0\].
To solve such equations, we often use the Quadratic Formula. This formula provides the solutions to any quadratic equation.

Identifying Coefficients

First, recognize the coefficients in your quadratic equation. In the equation \(4m^2 + m - 3 = 0\):

• The coefficient of \(m^2\) (which is \(a\)) is 4.
• The coefficient of \(m\) (which is \(b\)) is 1.
• The constant term (which is \(c\)) is -3.

Identifying these coefficients is crucial because they will be substituted into the Quadratic Formula.

Discriminant Calculation

The discriminant is found under the square root in the Quadratic Formula. It is calculated as:

\(b^2 - 4ac\).
In our example:

• \(b^2 = 1^2 = 1\).
• \(4ac = 4 \times 4 \times (-3) = -48\).
• The discriminant: \(1 - (-48) = 1 + 48 = 49\).

The discriminant helps us determine the number and type of solutions. A positive discriminant means two real and distinct solutions.

Square Root Simplification

After calculating the discriminant, the next step is to simplify the square root of the discriminant. For the example, the square root of 49 simplifies to:

• \(\root{49} = 7\)

Now, substitute back into the formula: \(m = \frac{-1 \pm 7}{8}\).

Splitting Solutions

Finally, split the equation into two separate solutions by considering both the plus and minus signs in the formula. For our example:

• First solution: \(m = \frac{-1 + 7}{8} = \frac{6}{8} = \frac{3}{4}\).
• Second solution: \(m = \frac{-1 - 7}{8} = \frac{-8}{8} = -1\).

Thus, the quadratic equation \(4m^2 + m - 3 = 0\) has two solutions: \(m = \frac{3}{4}\) and \m = -1\.