Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$3 x^{2}-6 x=4$$

Step-by-Step Solution

Answer

The real solutions are \(x = 1 + \frac{\sqrt{21}}{3}\) and \(x = 1 - \frac{\sqrt{21}}{3}\).

Step 1: Move All Terms to One Side

Start by moving all terms in the equation to one side to set the equation to zero. Subtract 4 from both sides:\[3x^2 - 6x - 4 = 0\]

Step 2: Identify Coefficients for the Quadratic Formula

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). For this equation:- \(a = 3\)- \(b = -6\)- \(c = -4\)

Step 3: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.Substitute the coefficients into the formula:\[x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \times 3 \times (-4)}}{2 \times 3}\]Simplify the expression:\[x = \frac{6 \pm \sqrt{36 + 48}}{6}\]\[x = \frac{6 \pm \sqrt{84}}{6}\]

Step 4: Simplify the Square Root

Calculate the square root of 84:\[\sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}\]Substitute back into the expression:\[x = \frac{6 \pm 2\sqrt{21}}{6}\]

Step 5: Simplify the Fraction

Simplify the expression:\[x = 1 \pm \frac{\sqrt{21}}{3}\]Thus, the solutions are:\[x = 1 + \frac{\sqrt{21}}{3}\] and \[x = 1 - \frac{\sqrt{21}}{3}\]

Step 6: Verify Graphically

To verify graphically, plot the quadratic function \(y = 3x^2 - 6x - 4\) and observe the points where the curve intersects the x-axis. The x-intercepts should match the solutions \(x = 1 + \frac{\sqrt{21}}{3}\) and \(x = 1 - \frac{\sqrt{21}}{3}\).

Key Concepts

Graphical Solutions

Graphical solutions are a visual way to understand quadratic equations. By plotting the equation on a coordinate plane, you can see where the parabola intersects the x-axis. The points where it meets the axis are known as the roots or solutions of the quadratic equation. When you graph the equation, you're essentially seeing a visual representation of all mathematical solutions. In our example, the equation \( y = 3x^2 - 6x - 4 \) is plotted as a parabola. This shape helps you identify the solutions by locating the x-intercepts, which are the values of \( x \) where \( y \) is zero.

• The x-intercepts are the solutions of the equation.
• The direction of the parabola (upwards or downwards) is determined by the leading coefficient (\( a \)). Here, it's positive, so it opens upwards.
• The vertex gives additional information about the maximum or minimum point of the parabola.

Graphical methods provide a powerful tool for verifying solutions and intuiting the behavior of quadratic equations.

Quadratic Formula

The quadratic formula is a crucial tool for finding the roots of a quadratic equation. It works for any equation of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Using the quadratic formula involves substituting the coefficients \( a \), \( b \), and \( c \) from the equation into the formula. This provides a straightforward way to find the roots:

• Compute \( b^2 - 4ac \), the discriminant, which indicates the nature of the roots:
• If positive, the equation has two distinct real roots.
• If zero, it has exactly one real root.
• If negative, it indicates no real roots, though complex ones may exist.

In the given problem, the quadratic formula led to a solution where \( x = 1 \pm \frac{\sqrt{21}}{3} \), simplifying our exploration of the roots. The quadratic formula is universal. It always works for any coefficients, making it a reliable method for solving quadratics.

Roots of Equations

The roots of a quadratic equation are the values of \( x \) that satisfy the equation. They happen where the parabola crosses the x-axis on a graph. These points are crucial because they're the solution to the equation; that's why quadratic solutions are also called roots or zeros.Finding the roots addresses the core question of the equation: For which \( x \) will the value of \( y \) (or \( f(x) \)) equal zero? In our equation \( 3x^2 - 6x - 4 = 0 \), we found:

• Two real and distinct roots since the discriminant was positive.
• The numerical values \( x = 1 + \frac{\sqrt{21}}{3} \) and \( x = 1 - \frac{\sqrt{21}}{3} \).

The calculation of these roots provides insight into how the equation behaves when graphed. With graphical solutions, the roots correspond visually to the x-intercepts of the parabola. Understanding the nature and value of these roots is crucial in solving and interpreting quadratic equations effectively.