Problem 69 - 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. $$-3 x^{2}+8 x=16$$

Step-by-Step Solution

Answer

The solutions of the quadratic equation \(-3x^2 +8x - 16 = 0\) are x = \(\frac{4 + 4\sqrt{2}i}{3}\) and x = \(\frac{4 - 4\sqrt{2}i}{3}\). The zeros of the function \(y = -3x^2 +8x - 16\) are the same as the solutions.

Step 1: Rewrite the equation in standard form

Rearrange the equation \(-3x^2 +8x = 16\) to standard form by subtracting 16 from both sides. This gives: \(-3x^2 +8x - 16 = 0\) . The coefficients are a=-3, b=8, and c=-16.

Step 2: Use the quadratic formula

Substitute the coefficients a, b, and c into the quadratic formula to find the roots. The roots or solutions are given by: \(x = \frac{-b\pm\sqrt{b^2 - 4ac }}{2a} = \frac{-8\pm\sqrt{(8)^2 - 4*(-3)*(-16) }}{2*(-3)}\).

Step 3: Simplify to find the roots

Calculate inside the square root first, then take the square root, and finally calculate the numerator and denominator separately. This yields: \(x = \frac{-8\pm\sqrt{64 - 192}}{-6} = \frac{-8\pm\sqrt{-128}}{-6}\). The term within the root is a negative number, indicating that the roots are complex numbers. Hence, \(x = \frac{-8\pm \sqrt{128}i}{-6} = \frac{8\mp 8\sqrt{2}i}{6} = \frac{4\mp 4\sqrt{2}i}{3}\). The roots of the equation are thus x = \(\frac{4 + 4\sqrt{2}i}{3}\) and x = \(\frac{4 - 4\sqrt{2}i}{3}\).

Step 4: Relate the solutions to the zeros of the quadratic function

In a quadratic function, the zeros refer to the x-values where the function equals zero. In this case, the zeros of the function \(y = -3x^2 +8x - 16\) would be the solutions of the equation, which are x = \(\frac{4 + 4\sqrt{2}i}{3}\) and x = \(\frac{4 - 4\sqrt{2}i}{3}\).

Key Concepts

Complex Roots

When we solve a quadratic equation and encounter a negative number inside the square root, this implies that the solutions are complex roots. Complex numbers consist of a real part and an imaginary part.
For imaginary numbers, we use the symbol 'i', which represents the square root of \(-1\). Therefore, when a quadratic equation has complex roots, they appear in the form \(a + bi\) and \(a - bi\). These are known as conjugates.

• The term inside the square root is called the discriminant.
• If the discriminant is negative, it indicates that the roots will be complex.

In our example, solving the equation with a discriminant of \(-128\) led us to the roots \(\frac{4 + 4\sqrt{2}i}{3}\) and \(\frac{4 - 4\sqrt{2}i}{3}\). These complex roots are conjugates, as is common in quadratic equations.

Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions of any quadratic equation, which has the general form \( ax^2 + bx + c = 0 \). The formula is written as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula is used to calculate the values of \(x\) that make the equation true.

• The quadratic formula accounts for all types of roots: real and complex.
• The key part is the discriminant, \( b^2 - 4ac \).

When the discriminant is positive, the equation has two distinct real roots. When it's zero, the equation has exactly one real root (or a repeated root). In cases where it's negative, like in our exercise, the equation yields complex roots. Using the quadratic formula allows you to systematically solve any quadratic equation, regardless of the discriminant's value.

Zeros of Quadratic Functions

In the context of quadratic functions, zeros refer to the \(x\)-values at which the function equals zero. These zeros are the solutions to the equation \(y = ax^2 + bx + c\).

• For the function to be zero, the \(y\) output must equal zero, leading to the equation \(ax^2 + bx + c = 0\).
• The solutions of this equation give us the zeros of the function.

In our example, the function \( y = -3x^2 + 8x - 16 \) was set equal to zero: \( -3x^2 + 8x - 16 = 0 \). The zeros found were the complex roots, \( \frac{4 + 4\sqrt{2}i}{3} \) and \( \frac{4 - 4\sqrt{2}i}{3} \).
These zeros tell us where the graph of the function intersects the \(x\)-axis, or more generally, where the output becomes zero. However, with complex zeros, the graph does not intersect the real \(x\)-axis and instead indicates an intricate behavior involving imaginary components, reflecting the roots’ complex nature.