Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ \frac{3}{5} x^{2}-x=\frac{1}{5} $$

Short answer

The solutions are \( x = \frac{5 + \sqrt{13}}{6} \) and \( x = \frac{5 - \sqrt{13}}{6} \).

Step-by-step solution

Write the equation in standard form

Start by moving all the terms to one side to set the equation to zero. \( \frac{3}{5}x^2 - x - \frac{1}{5} = 0 \)

Identify coefficients

Compare the equation \( \frac{3}{5}x^2 - x - \frac{1}{5} = 0 \) with the standard form \( ax^2 + bx + c = 0 \). Here, \( a = \frac{3}{5} \), \(b = -1 \), and \( c = -\frac{1}{5} \).

Apply the quadratic formula

The quadratic formula is given by: \( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \). Plug in the values of \( a \), \( b \), and \( c \): \( b^2 - 4ac = (-1)^2 - 4 \left( \frac{3}{5} \right) \left( -\frac{1}{5} \right) = 1 -\frac{12}{25} = \frac{25}{25} -\frac{12}{25} = \frac{13}{25} \). Now substitute into the quadratic formula: \( x = \frac{-(-1) \, \pm \, \sqrt{ \frac{13}{25} }}{2 \cdot \frac{3}{5}} = \frac{1 \, \pm \, \frac{\sqrt{13}}{5}}{\frac{6}{5}} = \frac{5 \left(1 \, \pm \, \frac{\sqrt{13}}{5} \right)}{6} = \frac {5 \pm \sqrt{13}}{6} \)

Simplify the expression

The solutions of the equation are: \( x = \frac{5 + \sqrt{13}}{6} \) and \( x = \frac{5 - \sqrt{13}}{6} \).

Key concepts

solving quadratic equations

Solving quadratic equations is a fundamental skill in algebra. These are equations of the form \(ax^2 + bx + c = 0\). There are several methods to solve them: factoring, completing the square, and using the quadratic formula. The quadratic formula is particularly useful because it works for any quadratic equation. The formula is: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This formula calculates the solutions, or roots, of the quadratic equation by considering the coefficients \(a\), \(b\), and \(c\).

It’s essential to set the equation to zero before using the quadratic formula. This ensures you are working in the proper form to apply the method. By following step-by-step procedures, you can systematically find the solutions, identifying whether they are real or complex.

standard form of a quadratic equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). In this form:

• \(a\) is the coefficient of \(x^2\) (quadratic term).
• \(b\) is the coefficient of \(x\) (linear term).
• \(c\) is the constant term.

This form is crucial because it standardizes the way we approach solving the equation.

For example, in the equation \(\frac{3}{5}x^2 - x = \frac{1}{5}\), you first need to bring all terms to one side to set the equation to zero: \(\frac{3}{5}x^2 - x - \frac{1}{5} = 0\). Now it is in standard form, comparable to \(ax^2 + bx + c\). Any quadratic equation should be written in this form before attempting to solve it.

real solutions

Real solutions to a quadratic equation occur when the solutions are real numbers and not complex or imaginary numbers. To determine if the solutions are real, you need to evaluate the discriminant, \(b^2 - 4ac\), which is the expression under the square root in the quadratic formula:

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

The discriminant indicates the nature of the roots:

• If \(b^2 - 4ac > 0\), there are two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution, meaning the two roots are equal.
• If \(b^2 - 4ac < 0\), the solutions are complex or imaginary, and hence, there are no real solutions.

In the example problem, the discriminant \( \frac{13}{25}\) is greater than zero, indicating that this equation has two distinct real solutions.

identifying coefficients

Identifying the coefficients \(a\), \(b\), and \(c\) in a quadratic equation is a crucial step in solving it. These coefficients are the numerical factors of the terms in the standard form \(ax^2 + bx + c = 0\). Here’s how you identify each:

• \(a\) is the coefficient of \(x^2\), the quadratic term.
• \(b\) is the coefficient of \(x\), the linear term.
• \(c\) is the constant term.

In the example \(\frac{3}{5}x^2 - x - \frac{1}{5} = 0\):

• \(a = \frac{3}{5}\)
• \(b = -1\)
• \(c = -\frac{1}{5}\)

Always compare your equation to the standard form to correctly identify these coefficients. Misidentifying coefficients can lead to incorrect solutions, so double-check your work!