Question

Mulriple Choice If \(b^{2}-4 a c>0,\) which conclusion can be made about the graph of \(f(x)=a x^{2}+b x+c, a \neq 0 ?\) (a) The graph has two distinct \(x\) -intercepts. (b) The graph has no \(x\) -intercepts. (c) The graph has three distinct \(x\) -intercepts. (d) The graph has one \(x\) -intercept.

Short answer

The graph has two distinct x-intercepts. (a).

Step-by-step solution

Understanding the Discriminant

The discriminant of a quadratic equation in the form of \(ax^2 + bx + c = 0\) is given by \(b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation.

Condition Given

The condition provided is \(b^2 - 4ac > 0\). This means the discriminant is positive.

Roots of the Quadratic Equation

When the discriminant \(b^2 - 4ac\) is greater than 0, the quadratic equation has two distinct real roots.

Graph Interpretation

Two distinct real roots mean that the graph of \(f(x) = ax^2 + bx + c\) intersects the x-axis at two distinct points.

Conclusion

The correct conclusion is that the graph has two distinct x-intercepts. Therefore, the answer is (a).

Key concepts

quadratic equation

A quadratic equation is a second-degree polynomial represented as \(ax^2 + bx + c = 0\), where \(a eq 0\). This type of equation forms a parabola when graphed on the coordinate plane. The general shape of this parabola can either open upwards or downwards, depending on whether the coefficient \(a\) is positive or negative.
One of the key features of a quadratic equation is its x-intercepts, which are the points where the graph crosses the x-axis. To find these x-intercepts, we need to solve the quadratic equation. This is often done using the quadratic formula, factoring, or by completing the square.

discriminant

The discriminant is a formula within the quadratic equation that helps evaluate the nature of the roots without actually solving the equation. It is given by the expression \(b^2 - 4ac\).
The value of the discriminant determines whether the quadratic equation has real roots and, if so, how many. Here’s what the different values mean:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), the equation has exactly one real root (also called a repeated or double root).
• If \(b^2 - 4ac < 0\), the equation has no real roots but two complex roots.

Understanding the discriminant is crucial for predicting the behavior of the quadratic graph, especially its x-intercepts.

x-intercepts

X-intercepts are the points where the graph of the quadratic equation crosses the x-axis. For the equation \(ax^2 + bx + c = 0\), these x-intercepts are the solutions, or roots, of the equation. Finding the x-intercepts involves solving the quadratic equation.
Depending on the value of the discriminant \(b^2 - 4ac\), we can predict the number of x-intercepts:

• If the discriminant is positive (\(b^2 - 4ac > 0\)), there are two distinct x-intercepts.
• If the discriminant is zero (\(b^2 - 4ac = 0\)), there is exactly one x-intercept.
• If the discriminant is negative (\(b^2 - 4ac < 0\)), there are no x-intercepts (the parabola does not cross the x-axis).

Identifying the x-intercepts helps us understand more about the solutions and overall shape of the quadratic graph.

real roots

Real roots are the solutions to the quadratic equation that are real numbers. These roots correspond to the points where the graph touches or crosses the x-axis. Real roots can be found using the quadratic formula, factoring, or other solving methods.
Based on the discriminant value:\(b^2 - 4ac\), we determine the type of real roots:

• Two distinct real roots: When \(b^2 - 4ac > 0\), there are two different x-values where the graph crosses the x-axis.
• One real root: When \(b^2 - 4ac = 0\), the graph just touches the x-axis at one point, indicating a repeated root.

Real roots are essential in understanding the behavior and graph of the quadratic equation. They provide valuable insight into how many times and where the parabola will intersect the x-axis, which affects many real-life applications.