Problem 90 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. (Lesson 9.5) $$ -3 x^{2}-x-4=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = -\frac{4}{3}\) and \(x = 1\).

Step 1: Setting up the quadratic equation

The first step is to set the equation \(-3x^{2} - x - 4 = 0\). This equation can be rewritten as \(-3x^{2} - x + 4 = 0\).

Step 2: Applying the quadratic formula

The quadratic formula is defined as \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where a, b and c are coefficients of the quadratic equation \(ax^{2} + bx + c\). For our equation, a = -3, b = -1, and c = 4. Therefore, substituting these values into the quadratic formula, we get \(x = \frac{-(-1) \pm \sqrt{(-1)^{2} - 4*-3*4}}{2*-3}\).

Step 3: Simplify the expressions

Simplify the above expression to get the roots of the quadratic equation. After solving, we get \(x = \frac{1 \pm \sqrt{1+48}}{-6} = \frac{1 \pm \sqrt{49}}{-6}\). This simplifies to \(x = \frac{1 \pm 7}{-6}\).

Step 4: Calculating the solutions

Solving for x gives the two solutions: \(x_{1}=\frac{1+7}{-6} = - \frac{8}{6} = - \frac{4}{3}\) and \(x_{2}=\frac{1-7}{-6} = \frac{6}{6} = 1\). These are the roots of the quadratic equation.

Step 5: Estimate solutions using the graph

Plot the equation \(-3x^{2} - x - 4 = 0\) using a graphing tool. The estimated solutions are the x-values where the curve crosses the x-axis. From the plotted graph, it should be observed that the curve crosses the x-axis at \(x_{1} = -\frac{4}{3}\) and \(x_{2} = 1\), which confirms the algebraic solutions.

Key Concepts

Graphing Quadratics

Graphing a quadratic equation can be a powerful visual tool to find its solutions, or roots. A quadratic equation is generally in the form of \(ax^2 + bx + c = 0\). To graph such an equation, you begin by constructing a parabola, which is a symmetric curve shaped like an arch. The axis of symmetry is a vertical line that goes through the vertex of the parabola, defined by \(x = -\frac{b}{2a}\).

When graphing, look for:

• Vertex: The highest or lowest point of the parabola.
• Intercepts: Points where the parabola crosses the axes; the x-intercepts are of particular interest as they represent the roots of the equation.
• Direction: The direction in which the parabola opens depends on the sign of \(a\). If \(a\) is positive, it opens upwards; if negative, it opens downwards.

By drawing the parabola on a graph, you can visually estimate the points where the curve intersects the x-axis, providing an approximation of the equation's solutions.

Quadratic Formula

The quadratic formula is a reliable method for finding the roots of any quadratic equation. It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows you to find the values of \(x\) that make the quadratic equation equal to zero, effectively finding its roots. Let's break down the steps:

• Identify coefficients: Recognize \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\).
• Calculate the discriminant: \(b^2 - 4ac\). This value inside the square root determines the nature of the roots.
• Apply the formula: Substitute the coefficients into the quadratic formula and solve for \(x\).

The quadratic formula is particularly useful when graphing doesn't provide precise intersections. It's a surefire algebraic method that confirms the visual estimate from a graph.

Roots of Quadratic Equations

Roots of a quadratic equation, also known as zeroes or solutions, are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Here’s how you can understand them:

• Nature of roots: Dependent on the discriminant \(b^2 - 4ac\):
  • If positive, the roots are real and different, and the parabola intersects the x-axis at two points.
  • If zero, there is exactly one real root, meaning the parabola just touches the x-axis.
  • If negative, the roots are complex, and the parabola does not intersect the x-axis.
• Finding roots: You can find the roots either graphically by locating where the parabola crosses the x-axis or algebraically using the quadratic formula or factoring (when possible).

Understanding these roots is crucial in algebra as they represent solutions to physical and mathematical problems modeled by quadratic equations.