Problem 107 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(3 u^{2}+7 u-2=0\)

Step-by-Step Solution

Answer

The solutions are \(u = \frac{-7 + \sqrt{73}}{6}\) and \(u = \frac{-7 - \sqrt{73}}{6}\).

Step 1: Identify the coefficients

In the quadratic equation \(3u^2 + 7u - 2 = 0\), identify the coefficients a, b, and c: \(a = 3\), \(b = 7\), \(c = -2\).

Step 2: Write down the quadratic formula

The quadratic formula to find the roots of \(ax^2 + bx + c = 0\) is given by: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Substitute the coefficients into the quadratic formula

Substitute \(a = 3\), \(b = 7\), and \(c = -2\) into the quadratic formula: \(u = \frac{-7 \pm \sqrt{7^2 - 4(3)(-2)}}{2(3)}\).

Step 4: Simplify inside the square root

Calculate inside the square root: \(u = \frac{-7 \pm \sqrt{49 + 24}}{6}\).

Step 5: Add the terms under the square root

Simplify under the square root: \(u = \frac{-7 \pm \sqrt{73}}{6}\).

Step 6: Find the two solutions

Write down the two possible solutions: \(u_1 = \frac{-7 + \sqrt{73}}{6}\) \(u_2 = \frac{-7 - \sqrt{73}}{6}\).

Key Concepts

Solving Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\) where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\).
The goal is to find the values of \(x\) that satisfy this equation.
One efficient method to solve quadratic equations is the Quadratic Formula, which is \(x = \frac{-b \, \pm \, \sqrt{b^2 \, - \, 4ac}}{2a}\).
This formula gives you the roots, or solutions, of any quadratic equation.
The process involves:

• Identifying the coefficients \(a\), \(b\), and \(c\)
• Substituting these coefficients into the Quadratic Formula
• Simplifying to find the roots

The formula is a powerful tool because it works for every quadratic equation, even when other methods may not.

Quadratic Coefficients

In the quadratic equation \(3u^2 + 7u - 2 = 0\), the numbers 3, 7, and -2 are the quadratic coefficients.
Specifically:

• \(a = 3\): the coefficient of \(u^2\)
• \(b = 7\): the coefficient of \(u\)
• \(c = -2\): the constant term

These coefficients are crucial for solving the equation using the Quadratic Formula.
They help define the shape and position of the parabola represented by the quadratic equation.
Knowing how to identify and substitute them correctly ensures accurate solutions.
Always ensure that \(a eq 0\); otherwise, the equation is not quadratic.

Roots of Quadratic Equations

The roots of a quadratic equation are the solutions for \(u\) when \(3u^2 + 7u - 2 = 0\) is satisfied.
These roots can be real or complex numbers.
Using the Quadratic Formula, the roots are given by:
\(u = \frac{-7 \pm \sqrt{7^2 - 4(3)(-2)}}{6}\)
This simplifies to:

• \(u_1 = \frac{-7 + \sqrt{73}}{6}\)
• \(u_2 = \frac{-7 - \sqrt{73}}{6}\)

Here, \(u_1\) and \(u_2\) are the two roots of the equation.
These roots can also be interpreted as the x-intercepts of the parabola represented by the quadratic equation.
Understanding how to find these roots is key to solving any quadratic equation.

Square Roots in Algebra

Square roots are fundamental in Algebra, especially in solving quadratic equations.
The symbol \( \,\sqrt{}\, \) represents the square root of a number.
For example, \( \sqrt{9} = 3 \) because 3 squared (\(3^2\)) is 9.
In the context of quadratic equations, the expression under the square root sign in the Quadratic Formula is called the discriminant (\(b^2 - 4ac\)).
For the equation \(3u^2 + 7u - 2 = 0\), the discriminant is \(49 + 24 = 73\).
The square root of 73 cannot be simplified further, so it is left as \( \sqrt{73}\).
This discriminant helps determine the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one real root (a repeated root).
• If \(b^2 - 4ac < 0\), the roots are complex numbers.

Knowing how to work with square roots is essential for correctly applying the Quadratic Formula.