Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ 6 x^{2}-x=2 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{2}{3}\) and \(x = -\frac{1}{2}\).

Step 1: Bring the Equation to Standard Form

We need to rearrange the quadratic equation into the standard form, which is \(ax^2 + bx + c = 0\). Start by subtracting 2 from both sides of the equation: \[6x^2 - x - 2 = 0\] Now the equation is in standard form with \(a = 6\), \(b = -1\), and \(c = -2\).

Step 2: Identify a, b, and c

Identify the coefficients in the quadratic equation \(6x^2 - x - 2 = 0\):- \(a = 6\)- \(b = -1\)- \(c = -2\)

Step 3: Write the Quadratic Formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]We will use this formula to solve for \(x\).

Step 4: Calculate the Discriminant

The discriminant, \(D\), is the part of the quadratic formula under the square root, calculated as \(b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into the discriminant:\[D = (-1)^2 - 4(6)(-2)\]\[D = 1 + 48 = 49\]

Step 5: Solve Using the Quadratic Formula

Substitute the values for \(a\), \(b\), and \(D\) into the quadratic formula:\[x = \frac{-(-1) \pm \sqrt{49}}{2 \times 6}\]\[x = \frac{1 \pm 7}{12}\]This will give us two possible solutions.

Step 6: Find the Two Solutions

Calculate the two possible values of \(x\):1. \(x_1 = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3}\)2. \(x_2 = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2}\)

Key Concepts

Standard Form of Quadratic Equation

A quadratic equation is a type of polynomial equation that involves terms up to the second degree. To solve quadratic equations using the quadratic formula, it is crucial to first convert it into its standard form. The standard form of a quadratic equation looks like this:

• \(ax^2 + bx + c = 0\)

Here, \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the variable. To transform any quadratic equation into this form, you may need to rearrange and combine terms. For our example, the equation \(6x^2 - x = 2\) should first be rearranged and simplified by moving all terms to one side:

• Subtract 2 from both sides to get: \(6x^2 - x - 2 = 0\)

Now, the equation is in standard form, ready for applying the quadratic formula. Identifying the coefficients, we have \(a = 6\), \(b = -1\), and \(c = -2\). Each coefficient plays a vital role in calculating the solutions for \(x\).
Understanding this step is essential as it sets the stage for the use of the quadratic formula accurately.

Discriminant in Quadratic Equations

The discriminant is a key part of the quadratic formula and helps determine the nature of the roots of the quadratic equation. It is found under the square root symbol in the quadratic formula:

• \(b^2 - 4ac\)

This value can give you insight into how many and what kind of solutions you should expect:

• If the discriminant is positive (\(> 0\)), there are two distinct real solutions.
• If the discriminant is zero (\(= 0\)), there is one real solution (also called a repeated or double root).
• If the discriminant is negative (\(< 0\)), the equation has no real solutions, but two complex solutions.

In our example \(6x^2 - x - 2 = 0\), the discriminant is calculated as follows:

• Substitute \(a = 6\), \(b = -1\), \(c = -2\) into the formula: \((-1)^2 - 4 \times 6 \times (-2) = 1 + 48 = 49\)

Since the discriminant is 49, a positive number, it indicates there are two distinct real solutions to this equation. Calculating the discriminant helps to anticipate the nature of the solutions, thereby guiding the use of the quadratic formula.

Solving Quadratic Equations

Once we have the standard form and the discriminant, we can use the quadratic formula to find the solutions of the quadratic equation. The quadratic formula is:

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

This formula allows us to solve for \(x\) directly by substituting the values of \(a\), \(b\), and the discriminant. Here's how you apply it to our example:

• Plug \(a = 6\), \(b = -1\), and the previously calculated \(discriminant = 49\) into the formula.
\[x = \frac{-(-1) \pm \sqrt{49}}{2 \times 6} = \frac{1 \pm 7}{12}\]
• This will yield the two solutions for \(x\). Calculate both:
• \(x_1 = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3}\)
• \(x_2 = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2}\)

You now have the two solutions for the quadratic equation \(6x^2 - x - 2 = 0\). Making use of the quadratic formula is a powerful method because it provides a universal way to solve any quadratic equation, as long as the arithmetic is carefully handled. By mastering this method, you can solve complex equations with ease and accuracy.