Problem 75 - College Algebra

Question

Solve each equation in Exercises \(73-98\) by the method of your choice. \(5 x^{2}+2=11 x\)

Step-by-Step Solution

Answer

The solutions to the equation \(5x^2 + 2 = 11x\) are \(x = 2\) and \(x = 0.2\)

Step 1: Rearrange the equation

Rearrange the equation \(5x^2 + 2 = 11x\) to bring it in the standard format to obtain \(5x^2 - 11x + 2 = 0\), where \(a = 5\), \(b = -11\), and \(c = 2\).

Step 2: Apply the Quadratic Formula

Substitute \(a = 5\), \(b = -11\), and \(c = 2\) into the quadratic formula \(-b ± sqrt(b^2 - 4ac) / 2a\). This results in the equation \(x = [11 ± sqrt((-11)^2 - 4*5*2)] / (2*5)\)

Step 3: Solve for x

Evaluate the expression in the quadratic formula to solve for \(x\). \(x = [11 ± sqrt(121 - 40)] / 10\), which simplifies to \(x = [11 ± sqrt(81)] / 10\). Hence, the two solutions are \(x = (11 + 9) / 10 = 2\) or \(x = (11 - 9) / 10 = 0.2\)

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool in mathematics used to find the roots of quadratic equations. Quadratic equations are those that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The formula provides a straightforward way to find solutions, or roots, for the equation. The formula is:

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

This fundamental formula considers the coefficients of the equation and allows for the calculation of both possible values of \(x\) that make the equation true. It includes terms for both the square root and the division, crucial in finding the solution.

Solving Quadratic Equations

Solving quadratic equations can be done using various methods, but the quadratic formula is one of the most reliable and universally applicable.

• Reformulate the equation into the standard quadratic form: \(ax^2 + bx + c = 0\).
• Identify the coefficients \(a\), \(b\), and \(c\).
• Plug these values into the quadratic formula.
• Simplify the expressions to solve for \(x\).

A key part of solving these equations involves calculating the discriminant: \(b^2 - 4ac\). This term reveals the nature of the roots. If it's positive, there are two distinct real roots; if zero, there's exactly one real root; if negative, the roots are complex and not real.

Algebraic Expressions

Algebraic expressions are a combination of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. They form the backbone of algebra and are essential in writing equations and inequalities.

• They consist of terms, which can be numbers, variables, or products of numbers and variables.
• When solving quadratic equations, understanding how to manipulate these expressions is crucial.
• Rearranging equations often requires combining like terms or factoring.

In the context of quadratic equations, transforming an expression into its standard form is vital before applying the quadratic formula. It involves skillfully moving terms and factoring, ensuring each side of the equation balances properly.