Problem 7 - Introductory Algebra for College Students

Question

Find the \(x\) -intercepts for the parabola whose equation is given. If the \(x\) -intercepts are irrational numbers, round your answers to the nearest tenth. $$y=-x^{2}+8 x-12$$

Step-by-Step Solution

Answer

The x-intercepts of the parabola \(y = -x^2 + 8x - 12\) are \(x = 2\) and \(x = 6\).

Step 1: Represent the Equation by Setting y Equal to Zero

First, we start with the given equation of the function, \(y = -x^{2} + 8x - 12\), and set y equal to zero. This gives us the equation: \(0 = -x^2 + 8x - 12\).

Step 2: Apply the Quadratic Formula

Now apply the quadratic formula, using \(a = -1\), \(b = 8\), and \(c = -12\). This gives us \(x = \frac{-8 \pm \sqrt{8^2 - 4(-1)(-12)}}{2(-1)}\).

Step 3: Simplify the Equation

Continue to simplify the equation, calculating what's under the root in the quadratic formula first. This leads to \(x = \frac{-8 \pm \sqrt{64 - 48}}{-2}\). Further simplifying and doing the subtraction under the square root gives: \(x = \frac{-8 \pm \sqrt{16}}{-2}\). Under the square root, 16 equals to 4 when taking the root. Plug in this value to the equation. Now, we have two possible solutions: \(x = \frac{-8 + 4}{-2}\) and \(x = \frac{-8 - 4}{-2}\). Finally, continue calculating to get \(x = 2\) or \(x = 6\).

Key Concepts

x-intercepts

The x-intercepts of a parabola are the points where the parabola intersects the x-axis. At these points, the value of y is zero. So, to find the x-intercepts, we set the equation of the parabola to zero and solve for x.

The equation given in the exercise is: \(y = -x^2 + 8x - 12\). To find the x-intercepts, we set \(y = 0\) to get the equation \(0 = -x^2 + 8x - 12\).

After solving for x, these solutions will give us the specific x-coordinates where the parabola meets the x-axis. These points are crucial as they often help in graphing the parabola or in understanding the function’s roots.

• The x-intercepts are the solutions to the equation when \(y = 0\).
• They provide the roots of the quadratic equation or points of intersection with the x-axis.
• For our given equation, the x-intercepts are found to be \(x = 2\) and \(x = 6\).

quadratic formula

The quadratic formula is a robust tool used to find the roots (x-intercepts) of a quadratic equation. A quadratic equation is generally in the form \(ax^2 + bx + c = 0\). The quadratic formula expresses the solutions for x as:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

For the given equation \(-x^2 + 8x - 12 = 0\), we identify the coefficients: \(a = -1\), \(b = 8\), and \(c = -12\). By substituting these into the quadratic formula, we can effectively solve for x.

• The part \(b^2 - 4ac\) under the square root is called the discriminant. It determines the nature of the roots.
• If the discriminant is positive, there are two real and distinct solutions. If it is zero, there is exactly one real solution. If it's negative, there are no real solutions.
• In our problem, the discriminant was \(16\), indicating two real solutions, and allowed to find the x-intercepts at \(x = 2\) and \(x = 6\).

solving parabolas

Solving a parabola involves finding points like vertices, intercepts, and sometimes the axis of symmetry. One common task is finding the x-intercepts by using either factoring, completing the square, or the quadratic formula.

When given a quadratic equation, setting y to zero can simplify it into the forms necessary for using these methods. For our problem, using the quadratic formula was particularly effective given the structure of the equation:

\(-x^2 + 8x - 12 = 0\).

• Begin by simplifying and rearranging the equation if necessary to best suit the chosen method.
• Using methods like factoring when applicable often offers simpler calculations but isn’t always possible.
• The quadratic formula is an all-encompassing method which works for any form of the equation as long as all necessary calculations are correctly performed.
• Obtained solutions, \(x = 2\) and \(x = 6\), help in plotting the parabola and understanding its graphical representation.

These steps not only provide the x-intercepts but assist in sketching and analyzing the parabola’s overall behavior on the graph.