Problem 38 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$7 x^{2}+12 x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -\frac{12}{7}\).

Step 1: Identify the Coefficients

The given quadratic equation is \(7x^2 + 12x = 0\). Identify the coefficients from the standard form \(ax^2 + bx + c = 0\). Here, \(a = 7\), \(b = 12\), and \(c = 0\).

Step 2: Apply the Quadratic Formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = 7\), \(b = 12\), and \(c = 0\) into the formula: \[ x = \frac{-12 \pm \sqrt{12^2 - 4 \times 7 \times 0}}{2 \times 7} \] Simplify to get: \[ x = \frac{-12 \pm \sqrt{144}}{14} \] \[ x = \frac{-12 \pm 12}{14} \]

Step 3: Solve for Both Roots

Calculate the values of \(x\) for both potential roots:1. \(x = \frac{-12 + 12}{14} = 0\)2. \(x = \frac{-12 - 12}{14} = \frac{-24}{14} = -\frac{12}{7}\)Thus, the solutions are \(x = 0\) and \(x = -\frac{12}{7}\).

Step 4: Check the Solution - Sum and Product of Roots

According to the sum of roots formula \(\alpha + \beta = -\frac{b}{a}\) and product \(\alpha \beta = \frac{c}{a}\), let's check:- Sum: \(0 + (-\frac{12}{7}) = -\frac{12}{7} = -\frac{12}{7} = -\frac{b}{a}\).- Product: \(0 \cdot (-\frac{12}{7}) = 0 = \frac{0}{7} = \frac{c}{a}\).The solutions satisfy both conditions.

Key Concepts

Quadratic Formula

The Quadratic Formula is a powerful tool for solving any quadratic equation of the form \(ax^2 + bx + c = 0\). This formula will give you the roots or solutions of the equation and is applicable no matter the value of the discriminant. The quadratic formula is:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In the formula, \(a\), \(b\), and \(c\) are the coefficients of the terms \(x^2\), \(x\), and the constant term, respectively. The symbol \(\pm\) indicates that there will generally be two solutions for \(x\), corresponding to the plus and minus.
To use the formula effectively, it is crucial to compute the discriminant \(b^2 - 4ac\) first, as it determines the nature of the roots:

• If \(b^2 - 4ac \gt 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root (a repeated root).
• If \(b^2 - 4ac \lt 0\), the roots are complex and not real.

Once the discriminant is calculated, you substitute it into the formula to find the roots. In our specific example, \(7x^2 + 12x = 0\), applying the quadratic formula leads us directly to finding \(x = 0\) and \(x = -\frac{12}{7}\).

Sum and Product of Roots

The sum and product of the roots of a quadratic equation are elegant relationships that help verify the correctness of obtained solutions. They arise directly from the coefficients of the quadratic equation \(ax^2 + bx + c = 0\).
The sum of the roots \((\alpha + \beta)\) is given by the equation:

• \(\alpha + \beta = -\frac{b}{a}\)

This formula tells us that the sum of the roots is equal to the negation of the coefficient of \(x\) divided by the coefficient of \(x^2\).
The product of the roots \((\alpha \beta)\) is calculated using:

• \(\alpha \beta = \frac{c}{a}\)

This shows that the product is the constant term \(c\) divided by the coefficient of \(x^2\), \(a\).
For this exercise, with the quadratic equation \(7x^2 + 12x = 0\), the roots \(x = 0\) and \(x = -\frac{12}{7}\) were found. We can verify them using:

• Sum: \(0 + (-\frac{12}{7}) = -\frac{12}{7}\), which matches \(-\frac{b}{a}\).
• Product: \(0 \cdot (-\frac{12}{7}) = 0\), fitting \(\frac{c}{a}\).

Both relationships check out, confirming our solutions are correct.

Solving Algebraic Equations

Solving algebraic equations, especially quadratic ones, involves finding values of the variable that make the equation true. Quadratic equations are a common type, and they typically have the form \(ax^2 + bx + c = 0\). Solving these requires a series of structured steps:
1. **Standard Form**: Initially, ensure the quadratic equation is in the standard form \(ax^2 + bx + c = 0\). This makes it easier to identify coefficients.
2. **Applying Formulas or Factoring**: Depending on the equation, you can solve the quadratic equation using either factoring, completing the square, or the quadratic formula. The choice depends on the equation's complexity and whether it can be factored easily.

• Factoring is quick but only applies when the quadratic is easily decomposable.
• The quadratic formula is universal—applicable in cases where factoring is not apparent.

3. **Verification**: Once the roots are calculated using any method, it's vital to verify them using either substitution back into the original equation or using the sum and product of roots as discussed.
In the given exercise, using the quadratic formula simplified finding the roots. These were then cross-verified using the sum and product of roots method, showcasing the importance of checking solutions.