Question

Find the x-intercepts of the graph of the equation. $$y=x^{2}+2 x+15$$

Short answer

The graph of the equation \(y=x^{2}+2x+15\) doesn't have any x-intercepts in the real number domain, as the solution to the associated quadratic equation results in complex roots.

Step-by-step solution

Set up the equation

To first find the x-intercepts, we will set our \(y\) value to zero and solve for \(x\). So our equation is: \(0=x^{2} + 2x + 15\).

Apply the quadratic formula

We will now solve for \(x\) by using the quadratic formula: \(x = [-b ± sqrt(b^{2}-4ac)] / 2a\). In this exercise, \(a = 1\), \(b = 2\), and \(c = 15\). Substituting these values into the quadratic formula gives us \(x = [-2 ± sqrt((2)^{2}-4*1*15)] / 2*1\).

Simplify the equation

The simplified version of the equation is \(x = [-2 ± sqrt(4-60)]/2\), which reduces further to: \(x = [-2 ± sqrt(-56)]/2\).

Solve for x

Given that sqrt(-56) is an imaginary number, we will end up with complex roots. Therefore the graph of the given equation has no x-intercepts in real number domain, since the discriminant is less than zero.

Key concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra for finding the roots of quadratic equations, which are of the form \(ax^2 + bx + c = 0\). This formula looks like this: \(x = \frac{{-b \pm \sqrt{{b^2-4ac}}}}{{2a}}\). The quadratic formula is incredibly useful because it always works for any quadratic equation, providing a direct method to find the x-intercepts of the graph, if they exist in the real number set.

When applied, the quadratic formula gives two solutions, corresponding to the points where the quadratic equation touches or crosses the x-axis. These solutions may be real and distinct, one real repeated solution, or complex numbers. As in our example with \(y = x^2 + 2x + 15\), the quadratic formula shows how you can solve for \(x\) by plugging in the values of \(a\), \(b\), and \(c\) from the equation. Steps include simplifying the expression under the square root (the discriminant), and then carrying out the operations to find the potential x-intercepts.

Complex Numbers

Understanding Complex Numbers

Complex numbers come into play when we are dealing with square roots of negative numbers. The standard form of a complex number is \(a + bi\), where \(a\) is the real part, \(b\) is the imaginary part, and \(i\) is the square root of -1, also called an imaginary unit. In the context of quadratic equations, if the discriminant \(b^2 - 4ac\) is negative, the solutions will be in terms of complex numbers, indicating that the quadratic doesn't cross the x-axis when graphed.

Returning to our example equation \(0 = x^2 + 2x + 15\), the solution revealed through the quadratic formula involves \(\sqrt{-56}\), which cannot be simplified in the realm of real numbers. This is where we interpret the square root of a negative number using complex numbers, introducing \(i\) to represent \(\sqrt{-1}\) and thus obtaining the complex roots of our quadratic equation.

Discriminant

The discriminant is a component within the quadratic formula \(\sqrt{b^2-4ac}\) that tells us the nature of the roots of a quadratic equation without actually calculating them. It is the expression under the square root sign and plays a crucial role in determining whether the quadratic equation will have two real and distinct solutions, one real solution (a repeated root), or complex solutions.

Here's a quick breakdown regarding the sign of the discriminant:

• If \(b^2 - 4ac > 0\), there are two distinct real roots.
• If \(b^2 - 4ac = 0\), there is one repeated real root.
• If \(b^2 - 4ac < 0\), there are two complex roots.

In the equation \(y = x^2 + 2x + 15\), the discriminant \(4 - 4*1*15\) is negative, hence we determined that this quadratic equation has no real solution for the x-intercepts and has complex roots instead.

Graphing Quadratics

Graphing quadratics is another visual method of finding x-intercepts. Every quadratic equation can be represented by a parabola on a coordinate plane. The points where the parabola intersects the x-axis are the x-intercepts, provided they exist. When graphing the quadratic equation \(y = ax^2 + bx + c\), the shape and direction of the parabola depend on the coefficient \(a\). If \(a > 0\), the parabola opens upwards; if \(a < 0\), it opens downwards.

The x-intercepts can be found by setting \(y = 0\) and solving the equation, which often involves factoring, completing the square, or using the quadratic formula. If the parabola does not touch or cross the x-axis, as in the case of our example with \(y = x^2 + 2x + 15\), the quadratic has no real x-intercepts, showcasing how the discriminant's sign and the graphing process are in agreement regarding the roots of a quadratic equation.