Problem 51 - Algebra A Combined Function

Question

Solve each equation. $$ 5 x^{2}-6 x-8=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -0.8\).

Step 1: Identify the Quadratic Equation

Recognize that the given equation \(5x^2 - 6x - 8 = 0\) is a quadratic equation in the standard form \(ax^2 + bx + c = 0\). Here, \(a = 5\), \(b = -6\), and \(c = -8\).

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions of the equation. Substitute \(a\), \(b\), and \(c\) into the formula.

Step 3: Calculate the Discriminant

Calculate the discriminant using the formula \(b^2 - 4ac\). Substitute \(b = -6\), \(a = 5\), and \(c = -8\):\[-6^2 - 4 \times 5 \times (-8) = 36 + 160 = 196.\]

Step 4: Compute the Square Root of the Discriminant

Since the discriminant is 196, compute the square root:\(\sqrt{196} = 14\).

Step 5: Solve for Both Roots of the Equation

Substitute the discriminant and other values into the quadratic formula:- For the plus sign: \[x = \frac{-(-6) + 14}{2 \times 5} = \frac{6 + 14}{10} = 2.\]- For the minus sign: \[x = \frac{-(-6) - 14}{2 \times 5} = \frac{6 - 14}{10} = -0.8.\]

Step 6: Conclude the Solutions

The solutions to the quadratic equation \(5x^2 - 6x - 8 = 0\) are \(x = 2\) and \(x = -0.8\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It provides the solutions, also known as roots, of the equation. The formula is expressed as:

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

Using this formula requires identifying the coefficients \( a \), \( b \), and \( c \) in the equation.
First, plug these coefficients into the formula, making sure to carefully substitute each value in the correct place to avoid calculation errors.
The "\( \pm \)" symbol indicates that there will generally be two solutions, corresponding to the plus and minus operations in the formula.

Discriminant

The discriminant is the part of the quadratic formula under the square root, represented by \( b^2 - 4ac \). This value tells us important information about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If it is negative, the equation has no real roots but two complex conjugate roots.

For example, in our exercise, the discriminant is \( 196 \), which is positive.
This means the quadratic equation has two distinct real roots.
This calculation is crucial as it influences the next steps in solving the equations, and preparing for root solving is essential.

Root Solving

Finding the roots of a quadratic equation is essentially solving for \( x \), where the quadratic expression equals zero. Using the quadratic formula, after calculating the discriminant, we proceed to find these roots:

• For the positive square root, substitute the values into \( x = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \).
• For the negative square root, use \( x = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \).

In our example problem, substituting these values gives us two solutions: \( x = 2 \) and \( x = -0.8 \).
These represent the points where the quadratic equation intersects the x-axis.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is always expressed as \( ax^2 + bx + c = 0 \). Recognizing this form is essential for using the quadratic formula effectively.

• The coefficient \( a \) is associated with \( x^2 \) and can never be zero since it would no longer be a quadratic equation.
• \( b \) is the coefficient of \( x \), and
• \( c \) is the constant term.

Identifying these coefficients properly is the first step in solving a quadratic equation.
This structure helps organize the path forward to root finding since all succeeding steps depend on accurately plugged coefficients.