Problem 55 - Algebra 1

Question

Find the x-intercepts of the graph of the equation. $$y=2 x^{2}-2 x-12$$

Step-by-Step Solution

Answer

The x-intercepts of the graph of the equation \(y=2 x^{2}-2 x-12\) are \(x = -2\) and \(x = 3\).

Step 1: Write down the equation

The first step involves noting down the equation given. The equation provided is \(y=2 x^{2}-2 x-12\). To find the x-intercepts, one must solve the equation for when y equals to 0, therefore we get \(2 x^{2}-2 x-12=0\).

Step 2: Rearrange the equation

Rearrange the equation in the form \(ax^{2}+bx+c=0\) where \(a=2\), \(b=-2\), and \(c=-12\). This is the standard form of a quadratic equation.

Step 3: Use the quadratic formula

Since we have a quadratic equation, we can use the quadratic formula to solve for the values of x when y equals to 0. The quadratic formula is \(x = [-b \pm \sqrt{(b^{2}-4ac)}]/(2a)\).

Step 4: Plugging the values

Next, we plug the values of a, b, and c into the formula. We get \(x = [2 \pm \sqrt{(-2)^{2}-4*2*(-12)}]/(2*2)\). This simplifies to \(x = [2 \pm \sqrt{4+96}]/4 = [2 \pm \sqrt{100}]/4 = [2 \pm 10]/4\).

Step 5: Solve for x

Finally, solve for x and we get two solution : \(x_1 = (2 + 10)/4 = 3\), and \(x_2 = (2 - 10)/4 = -2\)

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that represent a parabola in algebraic form. These equations take the general form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not equal to zero. The equation \(y=2x^2-2x-12\) is an example of a quadratic equation. When you are finding the x-intercepts of a parabola, you are essentially searching for the points where the graph crosses the x-axis. To do this, you set the y-value to zero and solve the resulting quadratic equation to find the values of x. This is a fundamental concept in algebra which finds use in various applications, including physics, engineering, and economics.

Quadratic Formula

The quadratic formula is used to solve quadratic equations and is derived from completing the square in the standard form of a quadratic equation. It is given by \(x = [-b \pm \sqrt{(b^2 - 4ac)}] / (2a)\). The symbols '\(\pm\)' indicate that there are typically two solutions to a quadratic equation, corresponding to the two x-intercepts of the graph. The term inside the square root, \(b^2 - 4ac\), is known as the discriminant. It determines the nature and number of solutions. If the discriminant is positive, there are two real and distinct solutions. A zero discriminant results in one real solution, while a negative discriminant indicates that there are no real solutions, only two complex solutions. Understanding and applying the quadratic formula is crucial for solving problems involving quadratic equations.

Graph of an Equation

The graph of an equation represents all the possible solutions of that equation on a coordinate plane. For quadratic equations, this graph is a parabola, which can open upwards or downwards depending on the sign of the coefficient \(a\). An important feature of the parabola is its intersection with the x-axis, which occurs at the x-intercepts. These x-intercepts can be easily visualized on the graph and are found by setting the y-value to zero and solving the equation for x. This visual representation assists in understanding the behavior of quadratic equations and how the coefficients \(a\), \(b\), and \(c\) affect the shape and position of the parabola.

Standard Form of a Quadratic Equation

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Any quadratic equation, regardless of how it is presented, can be rearranged into this form. Doing so is essential before applying the quadratic formula. The equation \(2x^2 - 2x - 12 = 0\) is a good example as it neatly fits the standard form with \(a=2\), \(b=-2\), and \(c=-12\). Once cast in this format, one can readily identify the coefficients \(a\), \(b\), and \(c\) and then use them in the quadratic formula to find the roots or x-intercepts of the equation. Recognizing and working with the standard form is a key skill in algebra that simplifies the process of solving quadratic equations.