Problem 67 - Precalculus : Building Concepts and Connections

Question

Find all solutions of the quadratic equation. Relate the solutions of the equation to the zeros of an appropriate quadratic function. $$5 x^{2}=-2 x-3$$

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(5x^{2}=-2x-3\), which are also the zeros of the function \(f(x) = 5x^{2}+2x+3\), are \(\frac{-1+\sqrt{14}i}{5}\) and \(\frac{-1-\sqrt{14}i}{5}\)

Step 1: Rearrange the equation

First, rearrange the given equation in the standard quadratic form \(ax^{2}+bx+c=0\). This would be \(5x^{2}+2x+3=0\)

Step 2: Apply Quadratic Formula

Next, apply the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) to find the solutions for \(x\). Plugging in the values, we get \[x=\frac{-2\pm\sqrt{(2)^{2}-4*5*3}}{2*5}\]

Step 3: Simplify the Solution

Now, simplify the expression which becomes \[x=\frac{-2\pm\sqrt{4-60}}{10} = \frac{-2\pm\sqrt{-56}}{10}\]. This indicates that we have complex roots as the number under the square root is negative.

Step 4: Find the real and imaginary parts

Extract the real and imaginary parts from the solution: \(x=\frac{-1\pm\sqrt{14}i}{5}\)

Step 5: Relate to zeros of the function

Finally, it can be stated that the zeros of the function \(f(x) = 5x^{2}+2x+3\) are \(\frac{-1+\sqrt{14}i}{5}\) and \(\frac{-1-\sqrt{14}i}{5}\). These values are the x-coordinates where the function crosses the x-axis, i.e., where \(f(x)=0\)

Key Concepts

Complex Roots

When solving quadratic equations, you might encounter complex roots. These occur when the discriminant (the expression under the square root in the quadratic formula) is negative.

In the equation \(5x^{2} + 2x + 3 = 0\), the discriminant is \(2^2 - 4 \cdot 5 \cdot 3\), which simplifies to \(-56\).

Because the discriminant is negative, the square root of a negative number leads to complex numbers, expressed using the imaginary unit \(i\), where \(i = \sqrt{-1}\).

Complex roots come in conjugate pairs, such as \(\frac{-1 + \sqrt{14}i}{5}\) and \(\frac{-1 - \sqrt{14}i}{5}\). These roots are not visible on the real number line but are crucial to solving equations fully.

Quadratic Formula

The quadratic formula is a powerful tool to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). It is given by:

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

This formula allows us to find the roots by substituting the values of \(a\), \(b\), and \(c\).

For the equation \(5x^2 + 2x + 3 = 0\), the coefficients are \(a = 5\), \(b = 2\), and \(c = 3\).

Plugging these into the quadratic formula, we get:

• \(x = \frac{-2 \pm \sqrt{4 - 60}}{10}\)
• Simplifying this yields complex roots, since the discriminant is negative.

The quadratic formula not only identifies roots but also indicates whether they are real or complex, based on the discriminant.

Zeros of a Function

In mathematics, zeros of a function, especially a quadratic function, are the values of \(x\) that make the function equal to zero.

For the function \(f(x) = 5x^2 + 2x + 3\), finding the zeros is equivalent to solving \(5x^2 + 2x + 3 = 0\).

When we find zeros, we're looking for the function's intersection points with the x-axis.

However, with complex roots like \(\frac{-1 + \sqrt{14}i}{5}\) and \(\frac{-1 - \sqrt{14}i}{5}\), these intersections are not with the real axis but with the complex plane.

This highlights that zeros can be complex, giving more depth to understanding how functions behave, even beyond the real number line.