Problem 115 - College Algebra Essentials

Question

In Exercises \(115-122,\) find all values of \(x\) satisfying the given conditions. $$ y=2 x^{2}-3 x \text { and } y=2 $$

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the equation are 1 and -0.5

Step 1: Set up the equation

We are given two equations \(y = 2 x^{2} - 3 x \) and \(y = 2 \), so we need to set these two equations equal to each other. This would give us \(2 x^{2} - 3 x = 2\).

Step 2: Rearrange the equation

Next, we need to rearrange this equation into a form where we can solve for \(x\). To do this, we will move 2 to the other side of equation by subtracting 2 from both sides, which gives us \(2 x^{2} - 3 x - 2 = 0\).

Step 3: Solve for \(x\)

To solve for \(x\), we can use the quadratic formula \(x = \frac{-b ± \sqrt{b^{2} - 4ac}}{2a}\). Here, \(a = 2\), \(b = -3\), and \(c = -2\). Substituting these values into the quadratic formula will give the roots or solutions for \(x\).

Step 4: Compute for roots

When we compute for \(x\), we get \(x = \frac{-(-3) ± \sqrt{(-3)^{2} - 4*2*(-2)}}{2*2}\), which simplifies to \(x = \frac{3 ± \sqrt{9 + 16}}{4}\), or \(x = \frac{3 ± \sqrt{25}}{4}\). After computing, we get \(x = 1\) and \(x = -0.5\)

Step 5: Conclusion

These calculations show that there are two values of \(x\) satisfying the given conditions: \(x = 1\) and \(x = -0.5\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool in algebra that allows you to find the roots of a quadratic equation, which is an equation in the form of \(ax^2 + bx + c = 0\). It's given by the following expression:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Here, the symbols \(a\), \(b\), and \(c\) represent the coefficients of the terms in the quadratic equation, with \(a\) being the coefficient of \(x^2\), \(b\) of \(x\), and \(c\) the constant term. The formula offers a straightforward method to derive the equation’s solutions, also called the roots or zeros, represented by the \(x\)-values for which the quadratic equation equals zero.

Applying the Formula

In the step-by-step solution, the quadratic formula was applied to the rearranged equation \(2x^2 - 3x - 2 = 0\). The values \(a = 2\), \(b = -3\), and \(c = -2\) were plugged into the formula, simplifying the square root of \(25\) to \(5\), and giving us two potential solutions for \(x\). This process is a universally applicable approach anytime you're faced with a quadratic equation.

Algebraic Equations

Algebraic equations are mathematical statements indicating that two expressions are equal. They can range from simple linear equations to more complex forms, such as quadratic equations. Solving these equations is crucial for finding unknown values that make the equation true.

Strategy for Solving

To solve algebraic equations, we typically perform operations to isolate the unknown variable on one side of the equation. With quadratic equations, like the one presented in the step-by-step solution, we often have to rearrange the equation to the standard form before applying solution methods like factoring, completing the square, or using the quadratic formula. The example shows how subtracting \(2\) from each side helps set the equation to a form amenable for the quadratic formula application.
Understanding the manipulation of algebraic expressions and mastering these strategies can help students solve a wide variety of problems in mathematics and applied areas.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions to the equation; that is, the values of \(x\) that make the equation's value zero. These roots can be real or complex and there can be one or two of them.

Understanding Roots

The discriminant, \(b^2 - 4ac\), within the quadratic formula provides insight into the nature of the roots. If the discriminant is positive, there are two distinct real roots, as showcased in our exercise. If it equals zero, there is one real root, and if it is negative, the roots are complex.
By solving \(2x^2 - 3x - 2 = 0\), we find two real roots, \(x = 1\) and \(x = -0.5\), indicating the points where the parabola representing the quadratic equation intersects the \(x\)-axis. This knowledge is not only fundamental to algebra but also critical for understanding graphs of functions and for applications in science and engineering contexts.