Problem 58 - Algebra 1

Question

Find the x-intercepts of the graph of the equation. $$y=x^{2}+7 x-2$$

Step-by-Step Solution

Answer

The x-intercepts of the graph of the equation \(y=x^{2}+7 x-2\) are \(x = \frac{-7 + \sqrt{57}}{2}\) and \(x = \frac{-7 - \sqrt{57}}{2}\)

Step 1: Setting the Equation to Zero

To find the x-intercepts, set the equation equal to zero: \(0 = x^{2} +7x -2\). This is because x-intercepts are the x-coordinates where the graph of a function intersects the x-axis, at which point \(y = 0\).

Step 2: Identify the coefficients of the quadratic equation

A general quadratic equation is given as \(ax^2 + bx + c = 0\). In this equation, \(a = 1\), \(b = 7\) and \(c = -2\).

Step 3: Apply the Quadratic Formula

Applying the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), substitute the identified values from Step 2. This gives \(x = \frac{-7 \pm \sqrt{7^2 - 4*1*(-2)}}{2*1}\). Simplify under the square root sign to obtain \(x = \frac{-7 \pm \sqrt{49 + 8}}{2}\).

Step 4: Final Calculation

Perform the final calculations to get the two solutions for \(x\): \(x = \frac{-7 \pm \sqrt{57}}{2}\). Therefore, the 2 solutions or x intercepts of the equation are \(x = \frac{-7 + \sqrt{57}}{2}\) and \(x = \frac{-7 - \sqrt{57}}{2}\).

Key Concepts

Quadratic Equation

When dealing with quadratic equations, you are essentially working with equations of the form \(ax^2 + bx + c = 0\). A quadratic equation is distinguished by the "squared" term, in this case, \(x^2\), and is a fundamental concept in algebra.

These types of equations often represent parabolas when plotted on a graph. The parabolic curve will intersect the x-axis at points called x-intercepts. Identifying these intercepts involves setting the quadratic equation equal to zero. This action determines the values of \(x\) where the curve touches the x-axis, hence making \(y = 0\).

In our exercise, the equation \(x^2 + 7x - 2\) is set to zero to find the x-intercepts. By this method, we transition from an expression for \(y\), which describes a parabola, to focusing on just \(x\) values where the graph crosses the x-axis.

Quadratic Formula

The quadratic formula is a critical tool for solving quadratic equations. It provides a straightforward way to find solutions for \(x\) in any equation of the form \(ax^2 + bx + c = 0\). The formula is written as:

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

Understanding this formula begins with recognizing the symbols involved:

• "\(b\)" and "\(c\)" are coefficients from the quadratic equation.
• The "\(\pm\)" symbol indicates there will usually be two solutions: one with addition and one with subtraction.
• The part under the square root, "\(b^2 - 4ac\)," is known as the discriminant and determines the number and nature of the solutions.

In our example, substituting "\(a = 1\)," "\(b = 7\)," and "\(c = -2\)" into the quadratic formula helps us arrive at the x-intercepts. The discriminant calculation \( \sqrt{49 + 8} = \sqrt{57} \) confirms the solutions are real and distinct. From here, calculation proceeds with: \( x = \frac{-7 \pm \sqrt{57}}{2} \).

Graphing

Graphing involves visually representing the solutions of equations, such as quadratic equations, on a coordinate system. For quadratic equations, graphing helps us understand the shape of the parabola, its vertex, direction of opening, and importantly, the x-intercepts.

Parabolas can open upwards or downwards based on the sign of the coefficient \(a\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards. In our scenario with \(y = x^2 + 7x - 2\), since \(a = 1\), it opens upwards.

The x-intercepts, which we found using the quadratic formula, mark the touching points of the graph on the x-axis, at \( \frac{-7 + \sqrt{57}}{2} \) and \( \frac{-7 - \sqrt{57}}{2} \). These reflect the solutions where \(y = 0\). Seeing these on a graph provides insight into the equation's behavior, illustrating both the parabola's tangible intersection points and the symmetry around its vertex.