Problem 51 - Elementary Algebra

Question

Use the quadratic formula to solve the following. $$x_{2}-0.6 x-0.27=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 0.9 \) and \( x = -0.3 \).

Step 1: Identify Coefficients

The quadratic equation is given as \( x^2 - 0.6x - 0.27 = 0 \). This means \( a = 1 \), \( b = -0.6 \), and \( c = -0.27 \).

Step 2: Write the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We'll use these coefficients to find the roots.

Step 3: Calculate the Discriminant

The discriminant \( \Delta \) is \( b^2 - 4ac \). Substituting the values gives \( (-0.6)^2 - 4 \times 1 \times (-0.27) \).

Step 4: Compute Discriminant Value

Calculate \( (-0.6)^2 = 0.36 \) and \( 4 \times 1 \times (-0.27) = -1.08 \). Therefore, \( \Delta = 0.36 + 1.08 = 1.44 \).

Step 5: Determine the Square Root of the Discriminant

\( \sqrt{1.44} = 1.2 \).

Step 6: Apply the Quadratic Formula

Substitute into \( x = \frac{-(-0.6) \pm 1.2}{2 \times 1} = \frac{0.6 \pm 1.2}{2} \).

Step 7: Find Each Solution

Calculate \( x_1 = \frac{0.6 + 1.2}{2} = \frac{1.8}{2} = 0.9 \) and \( x_2 = \frac{0.6 - 1.2}{2} = \frac{-0.6}{2} = -0.3 \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]It's an all-in-one solution that helps us find the roots of any quadratic equation by inputting the coefficients \(a\), \(b\), and \(c\), which are the numerical constants in the equation.
When using the quadratic formula:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Calculate the discriminant \(b^2 - 4ac\).
• Take the square root of the discriminant.
• Plug into the formula to find the two possible solutions for \(x\).

Discriminant

The discriminant is a special expression inside the quadratic formula given by \(b^2 - 4ac\). It plays a vital role in determining the nature of the roots of the quadratic equation.
Here's what different discriminant values tell us:

• If the discriminant is positive, the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two points.
• If the discriminant is zero, the quadratic equation has exactly one real root, also known as a repeated or double root. Here, the parabola touches the x-axis at just one point.
• If the discriminant is negative, there are no real roots, only complex roots. The parabola does not intersect the x-axis at all.

Understanding the value of the discriminant helps predict the number and type of solutions without actually solving the equation.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be found using the quadratic formula, as discussed earlier.

To find the roots:

• Apply the quadratic formula.
• Use the discriminant to understand the nature of the roots (real or complex).
• Calculate the exact values using arithmetic operations.

The roots could be real numbers, when they appear as intersections on the graph or complex numbers if they do not actually intersect the x-axis in real-number space. Understanding roots is essential, as they provide insights into graphing the equation and determining the factors of the quadratic function.