Question

Find the \(x\) -intercepts of the graph of the equation. $$y=5 x^{2}+8 x-8$$

Short answer

The x-intercepts of the graph of the equation \(y=5x^{2}+8x-8\) are \(x = -8/10 + \sqrt{224}/10\) and \(x = -8/10 - \sqrt{224}/10\).

Step-by-step solution

Set Up the Equation

Because we're finding the x-intercepts, this means that \(y = 0\). So replace \(y\) with 0 in the given equation: \(0 = 5x^{2} + 8x - 8\).

Apply the quadratic formula

The equation is now in the standard form of a quadratic equation \(ax^2 + bx + c = 0\). The quadratic formula to find the solutions (x-intercepts) is given by \(x = [-b \pm \sqrt{(b^2-4ac)}] / 2a\). In this case, \(a = 5\), \(b = 8\), and \(c = -8\). We plug these values into the quadratic formula to solve for \(x\).

Solve for \(x\)

After substituting \(a\), \(b\), and \(c\) into the quadratic formula, we get \(x = [-8 \pm \sqrt{(8^2-4*5*-8)}] / 2*5\), which simplifies to \(x = [-8 \pm \sqrt{64+160}] /10\). Further simplification gives \(x = [-8 \pm \sqrt{224}] / 10\), and thus the final solutions after simplifying the radical are \(x = -8/10 \pm \sqrt{224}/10\).

Key concepts

X-Intercepts

The x-intercepts of a graph are the points where the curve crosses the x-axis. In other words, they represent the values of x for which the graphed equation equals zero. Finding the x-intercepts can be particularly useful in understanding the shape and characteristics of quadratic graphs.

When dealing with quadratic equations, the x-intercepts are also referred to as the 'roots' or 'solutions' of the equation. For example, in the equation \(y = 5x^2 + 8x - 8\), if we want to find the x-intercepts, we need to solve for the values of x when \(y = 0\). This leads straight into the realm of solving quadratic equations, and the quadratic formula offers a systematic approach for this purpose.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The coefficients \(a\), \(b\), and \(c\) determine the shape and position of the parabola represented by the equation when graphed on a coordinate plane.

Quadratics are notable for their characteristic 'U' shape known as a parabola. Typically, quadratic equations can have either two real solutions, one real solution, or two complex solutions, depending on the nature of the discriminant \(b^2 - 4ac\), which is part of the quadratic formula.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of x that make the equation true. There are several methods to find the roots, such as factoring, completing the square, graphing, and using the quadratic formula. The quadratic formula, which is derived from the process of completing the square, provides a surefire way to find the solutions of any quadratic equation.

The formula is \(x = [-b \pm \sqrt{b^2 - 4ac}] / (2a)\), where \(\pm\) indicates that there will generally be two solutions, corresponding to the two x-intercepts of the graph of the quadratic function. It is universally applicable to all forms of quadratic equations as long as they are in standard form and is especially useful when the equation cannot be easily factored.

Standard Form of a Quadratic

The standard form of a quadratic equation is \(ax^{2} + bx + c = 0\). In this form, a, b, and c are specific numbers and a represents the coefficient of the squared term, which cannot be zero. If a were to be zero, the equation would no longer be quadratic but linear.

The standard form is especially important when applying the quadratic formula since the formula assumes that the equation is expressed in this way. It's also handy for identifying the direction in which the parabola opens: upward if a is positive, and downward if a is negative. Thus, when solving quadratic equations like \(5x^{2} + 8x - 8 = 0\), ensuring that it is in standard form is crucial for finding accurate solutions.