Problem 28 - Algebra 1: Concepts and Skills

Question

Use a graph to estimate the solutions of the equation. Check your solutions algebraically. $$x^{2}-x=12$$

Step-by-Step Solution

Answer

The algebraic solutions to the equation \(x^2 - x - 12 = 0\) are \(x = 4\) and \(x = -3\). Both solutions should verify when graphing the equation.

Step 1: Rewrite the Equation in Standard Form

The standard form of a quadratic equation is \(ax^{2}+bx+c=0\). Given equation is \(x^2-x=12\). To rewrite it in standard form, subtract 12 from each side to get \(x^2 - x - 12 = 0\).

Step 2: Plot the Graph of the Equation

The equation \(x^2 - x - 12 = 0\) represents a parabola. By graphing this equation, solutions can be estimated as the x-intercepts points where the graph intersects the x-axis. It is expected to have two points of intersection, corresponding to the two roots of a quadratic equation.

Step 3: Checking Solutions Algebraically

The solutions to the quadratic equation can be found using the quadratic formula \(\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). For this equation, \(a = 1\), \(b = -1\), and \(c = -12\). Substituting these values into the quadratic formula yields two solutions: \(x = 4\) and \(x = -3\).

Key Concepts

Graphing Quadratic Equations

Understanding how to graph quadratic equations is essential for visualizing the relationship between the variables in the equation. When graphing a quadratic equation, you are essentially plotting a parabola, a U-shaped curve. You can use various methods such as finding the vertex, the axis of symmetry, and plotting points to make the graph.

For the quadratic equation in standard form, \( x^2 - x - 12 = 0 \), you would identify the vertex and plot several points around it to create the shape of the parabola. Remember that the parabola opens upwards if the coefficient of \( x^2 \) is positive, and downwards if it is negative. In this case, since the coefficient is positive, we expect the parabola to open upwards. Estimating the solutions involves finding where the parabola crosses the x-axis, which are the points where the value of \( y \) is zero.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is a crucial concept in understanding the structure of these equations. It is expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \) is not equal to zero.

This form allows you to analyze the components of the parabola, like its direction, width, and the x-intercepts (solutions). For the given equation \( x^2 - x - 12 \), rewriting it in standard form shows us directly that \( a = 1\), \( b = -1\), and \( c = -12\), which will be necessary for further analysis such as using the quadratic formula or completing the square.

Quadratic Formula

The quadratic formula is a powerful tool for finding the exact solutions of a quadratic equation when they are not easily factored. The formula is given by \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients from the standard form of the quadratic equation.

It's a universally applicable method for solving any quadratic equation, offering both real and complex solutions. For our specific equation \( x^2 - x - 12 = 0 \), by substituting \( a = 1\), \( b = -1\), and \( c = -12\) into the formula, we obtain the precise solutions, which are \( x = 4\) and \( x = -3\). These solutions are the points where the parabola crosses the x-axis.

X-intercepts of a Parabola

The x-intercepts of a parabola, also known as the roots or zeros, are the points at which the parabola intersects the x-axis. These points are significant because they represent the solutions to the quadratic equation.

To find the x-intercepts algebraically, you can use factoring, completing the square, or the quadratic formula. For the graphed parabola of the equation \( x^2 - x - 12 = 0 \), we look for the points where \( y \) is zero. In our case, using the quadratic formula showed that the x-intercepts are \( x = 4\) and \( x = -3\), which means the parabola crosses the x-axis at those points. These intercepts provide a clear visual validation of the solutions found algebraically.