Problem 62 - Algebra 2

Question

Without graphing, tell how many \(x\) -intercepts each function has. $$ y=10 x^{2}+13 x-3 $$

Step-by-Step Solution

Answer

The function has two distinct x-intercepts.

Step 1: Determine the Discriminant

Calculate the discriminant of the quadratic equation using the formula: discriminant \(\Delta\) = \(b^2 - 4ac\). For the quadratic equation \(ax^2 + bx + c = 0\), \(a = 10\), \(b = 13\), and \(c = -3\).

Step 2: Calculate the Discriminant

Substitute the values into the discriminant formula to get \(\Delta = 13^2 - 4(10)(-3)\). Calculate the value to determine the number of \(x\)-intercepts.

Step 3: Interpret the Discriminant

If \(\Delta > 0\), the quadratic function has two distinct \(x\)-intercepts. If \(\Delta = 0\), there is one \(x\)-intercept (a repeated root). If \(\Delta < 0\), the quadratic function has no \(x\)-intercepts.

Key Concepts

Discriminant of a Quadratic Equation

Understanding the discriminant is essential for solving quadratic equations and analyzing their graphs. The discriminant is a component within the quadratic formula which provides crucial information about the nature of the solutions without actually solving the equation.

The discriminant, denoted as \( \Delta \), is given by the formula \( \Delta = b^2 - 4ac \) for a quadratic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \), represent the coefficients of the terms of the quadratic equation.

By simply evaluating the discriminant, one can determine:

• Whether the equation has real or complex solutions.
• How many different real solutions exist.
• Whether the graph of the equation will touch or intersect the x-axis.

The value of the discriminant impacts the x-intercepts directly. A positive \( \Delta \) indicates two x-intercepts, while a zero \( \Delta \) signifies one, and a negative \( \Delta \) means none, as the parabola does not cross the x-axis.

Quadratic Equation Intercepts

The intercepts of a quadratic equation, specifically the x-intercepts, are the points at which the graph of the equation crosses the x-axis.

Making sense of x-intercepts is crucial for graphing parabolas and understanding the solutions to quadratic equations. The x-intercepts correspond to the roots or solutions of the equation \( ax^2 + bx + c = 0 \) and can be calculated by setting \( y = 0 \) and solving the equation for \( x \).

Visualizing X-Intercepts

On a graph, the x-intercepts can be seen as the points where the curve touches or intersects the horizontal axis. If a quadratic equation has:

• Two x-intercepts, the parabola crosses the x-axis at two points.
• One x-intercept, it means the vertex of the parabola is on the x-axis, forming a tangent.
• No x-intercepts, the parabola opens either up or down without ever crossing the x-axis.

In the function \( y = 10 x^{2}+13 x-3 \), the number of x-intercepts is determined by the discriminant of the quadratic equation.

Calculating the Discriminant

To calculate the discriminant of a quadratic equation, you follow a straightforward process.

Step-by-Step Calculation

For the quadratic equation \( y = 10 x^{2}+13 x-3 \), the coefficients are \( a = 10 \), \( b = 13 \) and \( c = -3 \). Using these values, you calculate the discriminant \( \Delta \) by substituting into the formula resulting in \( \Delta = 13^2 - 4(10)(-3) = 169 + 120 = 289 \). With this positive value of \( \Delta \) calculated, you can now infer that the quadratic equation will have two distinct x-intercepts, reflecting where the graph crosses the x-axis.

The discriminant is not just a number but a powerful tool that provides insights into the solutions of a quadratic equation. Positive, negative, or zero values each tell a different story of the graph's interaction with the x-axis, enabling students to visualize the problem before even graphing it.