Problem 49 - 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. $$-6 x^{2}+2 x+1=0$$

Step-by-Step Solution

Answer

The solutions are \( \frac{1 + \sqrt{7}}{6} \) and \( \frac{1 - \sqrt{7}}{6} \).

Step 1: Identify the coefficients

First, write down the given quadratic equation in standard form, which is \( ax^2 + bx + c = 0 \). In this case, the equation is already in standard form: \(-6x^2 + 2x + 1 = 0\). The coefficients are: \(a = -6\), \(b = 2\), and \(c = 1\).

Step 2: Substitute into the quadratic formula

The quadratic formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values \(a = -6\), \(b = 2\), and \(c = 1\) into the formula: \( x = \frac{-2 \pm \sqrt{(2)^2 - 4(-6)(1)}}{2(-6)} \).

Step 3: Calculate the discriminant

Calculate the value of the discriminant, \( b^2 - 4ac \). Using the values \( b = 2\), \( a = -6\), and \( c = 1\), compute \( (2)^2 - 4(-6)(1) = 4 + 24 = 28 \).

Step 4: Solve for x

Now substitute the discriminant back into the quadratic formula. The solution is \( x = \frac{-2 \pm \sqrt{28}}{-12} \). Simplifying further, \( \sqrt{28} = 2\sqrt{7} \), so the solutions are \( x = \frac{-2 \pm 2\sqrt{7}}{-12} \). Simplify to get \( x = \frac{-1 \pm \sqrt{7}}{-6} \). Thus, the solutions are \( x = \frac{1 + \sqrt{7}}{6} \) and \( x = \frac{1 - \sqrt{7}}{6} \).

Step 5: Verify the sum and product

The sum of the roots \( \frac{1 + \sqrt{7}}{6} + \frac{1 - \sqrt{7}}{6} \) equals \( \frac{2}{6} = \frac{1}{3} \), which matches \( -\frac{b}{a} = \frac{2}{-6} = \frac{1}{3} \). The product of the roots \( \left(\frac{1 + \sqrt{7}}{6}\right)\left(\frac{1 - \sqrt{7}}{6}\right) = \frac{1^2 - (\sqrt{7})^2}{36} = \frac{-6}{36} = -\frac{1}{6} \), which matches \( \frac{c}{a} = \frac{1}{-6} = -\frac{1}{6} \). Both relationships are verified.

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \, eq 0 \). There are several methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Among these, the quadratic formula is a powerful tool as it provides a direct way to find the solutions, or roots, of any quadratic equation.

The quadratic formula is:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. The symbol \( \pm \) indicates that the quadratic equation can have two possible solutions. Substituting the coefficients of the given quadratic equation into the formula allows us to solve for \( x \) directly.

• First determine these coefficients from your specific equation.
• Substitute them into the formula.
• Calculate the value under the square root, known as the discriminant.
• Finally, solve for \( x \) to find the roots of the equation.

Discriminant in Quadratic Equations

The discriminant is a part of the quadratic formula, located under the square root sign: \( b^2 - 4ac \). It plays a crucial role in determining the nature of the roots of the quadratic equation. By calculating the discriminant, you can determine:

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant is zero, the quadratic equation has exactly one real root, known as a repeated or double root.
• If the discriminant is negative, the quadratic equation has no real roots and instead has two complex roots.

In our exercise, we calculated the discriminant for \( -6x^2 + 2x + 1 = 0 \) to be 28, which is positive. This tells us that there are two distinct real roots for this equation. Calculating and interpreting the discriminant is a simple yet powerful step in solving quadratic equations.

Verification of Solutions

Once you have obtained the solutions to a quadratic equation, it's important to verify that these are correct. A common method is using the sum and product of the roots properties. These state that for a quadratic equation \( ax^2 + bx + c = 0 \):

• The sum of the roots \( x_1 + x_2 \) should equal \( -\frac{b}{a} \).
• The product of the roots \( x_1 \cdot x_2 \) should equal \( \frac{c}{a} \).

For our quadratic equation, the calculated roots are \( \frac{1 + \sqrt{7}}{6} \) and \( \frac{1 - \sqrt{7}}{6} \).

When their sum is calculated, we find:

\( \frac{1 + \sqrt{7}}{6} + \frac{1 - \sqrt{7}}{6} = \frac{2}{6} = \frac{1}{3} \), which matches the expected \( -\frac{b}{a} = \frac{1}{3} \).

Similarly, the product calculated is:

\( \left(\frac{1 + \sqrt{7}}{6} \right) \left(\frac{1 - \sqrt{7}}{6} \right) = -\frac{1}{6} \), matching \( \frac{c}{a} = -\frac{1}{6} \).

Confirming both the sum and product with the quadratic equation ensures that the solutions are indeed correct.