Problem 56 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(26 x=8 x^{2}+15\)

Step-by-Step Solution

Answer

The solutions of the equation are \(x = 2\) and \(x = \frac{15}{8}\).

Step 1: Rearrange the Equation

Rearranging the equation to the standard form gives \(8x^2 - 26x + 15 = 0\).

Step 2: Calculate the Discriminant

The discriminant can be calculated using the formula \(b^2 - 4ac\), which in this case is \((-26)^2 - 4*8*15 = 676 - 480 = 196\).

Step 3: Use the Quadratic Formula to Solve for \(x\)

The quadratic formula is \(x = \frac{-b \pm \sqrt{discriminant}}{2a}\). Filling in the values we found earlier gives two potential solutions for \(x\): \(x_1 = \frac{26 + \sqrt{196}}{2*8} = 2\) and \(x_2 = \frac{26 - \sqrt{196}}{2*8} = \frac{15}{8}\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. It provides a straightforward way to solve any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
By using the quadratic formula, you can find out where the equation equals zero, which gives the solutions or "roots" of the equation. This formula simplifies the process of solving quadratics, especially when factoring is complex or not possible.
To apply the quadratic formula:

• Identify the coefficients: \( a \), \( b \), and \( c \).
• Calculate the discriminant \( b^2 - 4ac \).
• Substitute \( a \), \( b \), and the discriminant into the formula.

This will yield the potential solutions for \( x \). Remember, the '±' indicates that there are usually two solutions, one for addition and one for subtraction.

Discriminant Calculation

The discriminant is a component of the quadratic formula that can tell us a lot about the solutions of the quadratic equation. The discriminant is calculated using the expression:
\[b^2 - 4ac.\]
The value of the discriminant determines the nature of the roots of the quadratic equation:

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

Understanding the discriminant is helpful because it gives you insight into how many solutions you can expect before you even solve the equation using the quadratic formula.

Standard Form of Quadratic Equation

To effectively solve a quadratic equation, it must be presented in its standard form. The standard form of a quadratic equation is:\[ax^2 + bx + c = 0,\]where:

• \(a\): is the coefficient of \(x^2\), and it should not be zero.
• \(b\): is the coefficient of \(x\), the linear term.
• \(c\): is the constant term or the term with no \(x\) attached.

Writing an equation in standard form is the first key step in solving it, whether by factoring, completing the square, or using the quadratic formula. In practice, you might need to rearrange your equation by moving all terms to one side and setting the equation to zero. Having it in this form makes it easier to identify \(a\), \(b\), and \(c\) when applying the quadratic formula or performing other solution methods.