Problem 86 - College Algebra

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$5 x^{2}=6-13 x$$

Step-by-Step Solution

Answer

The solution to the equation \(5x^{2}=6-13x\) is \( x=2 \) and \( x=0.6 \).

Step 1: Identify a, b and c in the equation

The quadratic equation is in the format \( ax^{2} + bx + c = 0 \). Here \( a=5 \), \( b=-13 \) and \( c=6 \)

Step 2: Apply the Quadratic Formula

The values should be placed into the quadratic formula: \( x=\frac{-(-13)\pm \sqrt{(-13)^{2} - 4 *5 *6}}{2 *(5)} \)

Step 3: Simplify the expression

On simplifying the expression, you obtain \( x=\frac{13\pm \sqrt{169-120}}{10} \rightarrow x=\frac{13\pm \sqrt{49}}{10} \rightarrow x=\frac{13\pm 7}{10} \)

Step 4: Calculate the solutions/roots

This gives you two roots: \( x=\frac{13+7}{10} = 2 \) and \( x=\frac{13-7}{10} = 0.6 \)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are any polynomial equations of the form \( ax^2 + bx + c = 0 \). This formula provides a straightforward way to find the solutions, or "roots", for these types of equations. The quadratic formula is:

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

Here, \( a \), \( b \), and \( c \) are constants corresponding to the coefficients of the terms in the quadratic equation. The symbol "\( \pm \)" implies that there will generally be two solutions for \( x \).
These solutions have different values depending on whether you add or subtract the square root term. Knowing the quadratic formula is essential for solving equations where rearranging or factoring isn't possible. It's important to substitute the correct values carefully to ensure accurate results.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Mathematically, these roots are where the graph of the quadratic function, typically a parabola, intersects the x-axis.

• If the discriminant \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If the discriminant \( b^2 - 4ac = 0 \), there is exactly one real root, also known as a repeated root.
• If the discriminant \( b^2 - 4ac < 0 \), there are no real roots; instead, the roots are complex numbers.

Understanding these scenarios helps in predicting the nature of the solutions before even calculating them. Finding the roots is crucial because it reveals vital information about the function, such as its intercepts with the x-axis.

Solving Quadratic Equations

Solving quadratic equations can be done using various methods, including factoring, completing the square, and using the quadratic formula. The choice of method often depends on the specific form and coefficients of the equation.

• **Factoring**: Simplifies the equation by expressing it as a product of linear factors. This method works best when the equation can be easily decomposed into integers.
• **Completing the Square**: Transforms the equation into a perfect square trinomial, facilitating easier solving by taking square roots.
• **Using the Quadratic Formula**: Applies to any quadratic equation and is particularly useful when factoring is not straightforward.

For the equation \( 5x^2 - 13x - 6 = 0 \), using the quadratic formula is optimal. By plugging in the values \( a=5 \), \( b=-13 \), and \( c=6 \) into the formula, you can efficiently calculate the exact roots of the equation. This helps in understanding the solutions and applying them to real-world problems.