Problem 122 - College Algebra Essentials

Question

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ \begin{aligned} &y_{1}=-x^{2}+4 x-2, y_{2}=-3 x^{2}+x-1, \text { and }\\\ &y_{1}-y_{2}=0 \end{aligned} $$

Step-by-Step Solution

Answer

The values of \(x\) for which \(y_1 - y_2 = 0\) are \(x = 0.5\) and \(x = -1\)

Step 1: Write down the equality

Write down the equality \(y_1 - y_2 = 0\). Substitute the given expressions for \(y_1\) and \(y_2\) which results in the equation: \(-x^2 + 4x -2 - (-3x^2 + x -1) = 0\)

Step 2: Simplify the equation

Simplify the equation by removing the brackets and combining like terms. This results in the quadratic equation \(2x^2 + 3x - 1 = 0\)

Step 3: Solve the quadratic equation

Solve the quadratic equation \(2x^2 + 3x - 1 = 0\). One common method is to use the quadratic formula: \[x = \frac{-b \pm sqrt{b^{2} - 4ac}}{2a},\]where \(a=2\), \(b=3\), and \(c=-1\)

Step 4: Calculate roots

Substitute the values \(a=2\), \(b=3\), and \(c=-1\) into the quadratic formula to calculate the roots. It results into two solutions:\[x_1 = \frac{-3 + sqrt{3^{2} - 4 * 2 * (-1)}}{2 * 2} = 0.5\]and\[x_2 = \frac{-3 - sqrt{3^{2} - 4 * 2 * (-1)}}{2 * 2} = -1\]

Key Concepts

Quadratic Formula

The quadratic formula is a versatile tool used to find the solutions, or roots, of a quadratic equation. A quadratic equation is typically in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This equation calculates the values of \(x\) that make the quadratic equation true. The symbol \(\pm\) indicates that there will generally be two solutions: one for the plus and one for the minus. The expression under the square root \(b^2 - 4ac\) is known as the discriminant, and it determines the nature of the roots.
The quadratic formula is essential for solving quadratics that don't factor nicely or when simplifying the problem manually is complex.

Roots of Equations

Roots of an equation are the values of \(x\) which satisfy the equation, meaning they make the equation equal to zero. In simple terms, roots can be thought of as the x-coordinates where the graph of the equation crosses the x-axis. Depending on the discriminant \(b^2 - 4ac\), a quadratic equation can have:

• Two distinct real roots (discriminant > 0),
• One real root (discriminant = 0), which is also known as a repeated or double root,
• No real roots and instead two complex roots (discriminant < 0).

In our solution, because the discriminant was positive, we found two distinct real roots \(x_1 = 0.5\) and \(x_2 = -1\). Each root satisfies the quadratic equation \(2x^2 + 3x - 1 = 0\), meaning it makes the equation equal to zero when substituted for \(x\).

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that make the equation true, bringing it to the form \(ax^2 + bx + c = 0\). There are different methods to solve such equations:

• Factoring: Splitting the equation into a product of simpler binomials, works well when the factors are easily identifiable.
• Completing the Square: A method converting the equation into a perfect square trinomial.
• Using the Quadratic Formula: This method works universally regardless of whether the equation can be factored.

For the provided exercise, using the quadratic formula was suitable because it provided a straightforward way to find the roots. By substituting the coefficients \(a\), \(b\), and \(c\) into the formula, we calculated the roots of the equation efficiently. This guarantees that we find all possible real and complex roots, making it a powerful method for any quadratic equation.