Problem 76 - College Algebra

Question

Solve each equation in Exercises \(73-98\) by the method of your choice. \(5 x^{2}=6-13 x\)

Step-by-Step Solution

Answer

The solution to the equation \(5x^2 = 6 - 13x\) are \(x = 0.4\) and \(x = -3\)

Step 1: Rearrange the Equation

Rearrange the equation in the standard form. Thus, move all terms to one side: \(5x^2 +13x - 6=0\)

Step 2: Identify coefficients

Identify the coefficients for the equation which are \(a = 5\), \(b = 13\) and \(c = -6\)

Step 3: Use the Quadratic Formula

Plug the coefficients into the quadratic formula which is \(x = [-b ± sqrt(b^2 - 4ac) ] / (2a) \) becoming \(x = [-13 ± sqrt((13)^2 - 4*5*(-6)) ] / (2*5)\)

Step 4: Simplify

Calculate the solutions to get \(x = [-13± sqrt(169+120)] / 10\) which further simplifies to \(x = -13±sqrt(289) / 10\) or \(x = -13±17 / 10\)

Step 5: Solve for x

The two solutions for x are \(x = (-13+17) / 10\) and \(x = (-13-17) / 10\), hence \(x = 0.4\), \(-3\)

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). It's expressed as:

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

This formula helps calculate the values of \(x\) that make the equation true. The expressions \(\pm\) indicate there are generally two solutions. These solutions arise because taking the square root can yield both a positive and negative result.
The term under the square root, \(b^2 - 4ac\), is the discriminant. It gives us insight into the nature of the roots:

• If it's positive, there are two distinct real solutions.
• If zero, one real solution (repeated root).
• If negative, two complex solutions.

Using the quadratic formula can be straightforward once you’ve identified the coefficients \(a\), \(b\), and \(c\) from the equation.

Solving Equations

Solving a quadratic equation involves finding the roots or solutions where the equation equals zero. To solve, follow these general steps:

• Rewrite the equation: First, ensure the equation is in standard quadratic form \(ax^2 + bx + c = 0\).
• Identify coefficients: Determine \(a\), \(b\), and \(c\) which are usually straightforward numbers.
• Apply quadratic formula: Substitute these values into the quadratic formula.
• Calculate roots: Simplify the resulting expression to find the roots.

Practicing these steps with various quadratic scenarios will increase familiarity and efficiency in solving such equations. Remember, correct algebraic manipulation is crucial in deriving the right solutions.

Coefficients in Quadratics

Coefficients in a quadratic equation are the numerical or constant parts that multiply the variable terms. In the equation \(ax^2 + bx + c = 0\):

• \(a\): Known as the leading coefficient, it multiplies \(x^2\).
• \(b\): This coefficient multiplies the \(x\) term.
• \(c\): The constant term has no variable attached.

These coefficients are crucial for solving quadratic equations, as they need to be correctly identified to use the quadratic formula. A small mistake in recognizing any of these can lead to wrong solutions.
Understanding their roles helps not only in solving but also in predicting the shape and position of the parabola when the equation is graphed.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This format is important because it sets a consistent structure for manipulating and solving quadratics. All quadratic equations should be rearranged or simplified to this form before solving.
When writing an equation in standard form:

• All terms should be moved to one side of the equation, resulting in zero on the other side.
• The terms are ordered by the power of \(x\), starting with \(x^2\), followed by \(x\), and then the constant \(c\).
• Ensure all coefficients are correct and grouped appropriately.

Expressing the equation in this manner is a critical step in using any method to solve quadratic equations, such as factoring, completing the square, or the quadratic formula. Keeping it neat and clear helps avoid errors in further calculations.