Problem 5 - A Graphical Approach to College Algebra

Question

Solve each problem. Do not use a calculator. $$\text { Solve }-4 x^{2}+5 x=1$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{4}\) and \(x = 1\).

Step 1: Set the Equation to Zero

Start by rearranging the equation \,\(-4x^2 + 5x = 1\)\, into a standard quadratic equation form, which is \,\(ax^2 + bx + c = 0\). To do this, subtract \(1\) from both sides to obtain: \,\(-4x^2 + 5x - 1 = 0\).

Step 2: Identify Coefficients

Identify the coefficients from the quadratic equation \(-4x^2 + 5x - 1 = 0\). Here, \(a = -4\), \(b = 5\), and \(c = -1\).

Step 3: Use the Quadratic Formula

The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute \(a = -4\), \(b = 5\), and \(c = -1\) into the formula to find the solutions for \(x\).

Step 4: Calculate the Discriminant

First, calculate the discriminant \(b^2 - 4ac\). Substitute the values: \(\text{Discriminant} = 5^2 - 4(-4)(-1) = 25 - 16 = 9\). Since the discriminant is positive, there are two real solutions.

Step 5: Solve using the Quadratic Formula

Substitute the discriminant back into the quadratic formula: \[x = \frac{-5 \pm \sqrt{9}}{2(-4)}\] Now solve for each scenario of \(\pm\).

Step 6: Solve for \(x_1\)

Calculate \(x_1\) using the plus sign: \[x_1 = \frac{-5 + \sqrt{9}}{-8} = \frac{-5 + 3}{-8} = \frac{-2}{-8} = \frac{1}{4}\] Thus, \(x_1 = \frac{1}{4}\).

Step 7: Solve for \(x_2\)

Calculate \(x_2\) using the minus sign: \[x_2 = \frac{-5 - \sqrt{9}}{-8} = \frac{-5 - 3}{-8} = \frac{-8}{-8} = 1\] Thus, \(x_2 = 1\).

Key Concepts

Quadratic Formula

The quadratic formula is a reliable method for solving any quadratic equation, which is in the form of \(ax^2 + bx + c = 0\). This formula can be very helpful, especially when other methods like factoring are not possible. The formula is as follows: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] It allows you to find the values of \(x\) that satisfy the equation. You simply plug in the coefficients \(a\), \(b\), and \(c\) from the quadratic equation into the formula.

• \(b\): the coefficient of \(x\)
• \(a\): the coefficient of \(x^2\)
• \(c\): the constant term

The beauty of the quadratic formula is its versatility. It works for any quadratic equation, no matter the complexity. Start by ensuring your equation is in standard form, identify coefficients, then substitute these into the formula. Solving the equation becomes straightforward as the formula will yield the solutions for \(x\).

Discriminant

The discriminant plays a key role in determining the nature of the solutions of a quadratic equation. It is the part of the quadratic formula under the square root symbol: \(b^2 - 4ac\). Calculating the discriminant is a valuable step because it tells us about the number and type of solutions without solving the equation fully. Here’s how it works:

• If the discriminant is positive (\(b^2 - 4ac > 0\)): you will have two distinct real solutions. The parabola crosses the x-axis at two points.
• If the discriminant is zero (\(b^2 - 4ac = 0\)): there is exactly one real solution. The parabola touches the x-axis at one point.
• If the discriminant is negative (\(b^2 - 4ac < 0\)): there are no real solutions. The solutions are complex or imaginary, meaning the parabola does not touch the x-axis.

Understanding the discriminant helps you anticipate the number of solutions and their nature even before computing them through the quadratic formula.

Real Solutions

Real solutions are solutions to a quadratic equation that can be plotted on a graph as points where the parabola touches or crosses the x-axis. Recognizing if the solutions of an equation are real or not can greatly impact how you interpret the findings of any problem involving quadratic equations.

• Two Real Solutions: Occurs when the discriminant is positive. Each solution corresponds to a point where the parabola intersects the x-axis.
• One Real Solution: Happens when the discriminant is zero, indicating the parabola just touches the x-axis.
• No Real Solutions: A negative discriminant means no intersection with the x-axis; solutions are complex numbers.

By using the quadratic formula and analyzing the discriminant, you can determine not just the values of \(x\) that solve the equation, but also visualize how the graph of the quadratic equation behaves on a Cartesian plane. This informs us if and where the parabola will interact with the horizontal line representing real numbers.