Problem 59 - Precalculus : Building Concepts and Connections

Question

Compute the zeros of the quadratic function. $$h(x)=3 x^{2}+8 x-16$$

Step-by-Step Solution

Answer

The zeros of the function \(h(x)=3x^2+8x-16\) are \(x = 4/3\) and \(x = -4\).

Step 1: Identify Coefficients

From the quadratic function \(h(x)=3x^2+8x-16\), identify that \(a=3\), \(b=8\), and \(c=-16\).

Step 2: Input Coefficients into Quadratic Formula

Input the coefficients into the quadratic formula \([-b±\sqrt{b^2-4ac}]/(2a)\). This gives us \([-8±\sqrt{8^2-4*3*(-16)}]/(2*3)\).

Step 3: Simplify under Square Root Sign

Simplify the equation under the square root sign: \(64+192\). This simplifies to \[-8±\sqrt{256}/6\].

Step 4: Calculate Square Root and Finalize Calculation

After finding the square root, which is 16, recalculate the formula to get two solutions, \(-8+16/6 = 4/3\) and \(-8-16/6 = -4\). These are the zeros of the function.

Key Concepts

Quadratic Formula

The quadratic formula is a mathematical formula used to find the solutions of a quadratic equation, which is any equation that can be rewritten in the form \(ax^2 + bx + c = 0\). The formula helps us determine the values of \(x\), known as the zeros or roots of the equation. It is expressed as:

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

This formula requires three key components from the quadratic equation: \(a\), \(b\), and \(c\), which are the coefficients of \(x^2\), \(x\), and the constant term respectively.
When you apply the quadratic formula, the "±" symbol indicates that there will potentially be two solutions for \(x\). One solution is obtained by using the plus sign, and the other by using the minus sign. This is because quadratic equations, when plotted as graphs, typically conform to a parabolic shape that intersects the x-axis up to two times, representing the zeros of the function.

Zeros of Quadratic Function

The zeros of a quadratic function are the x-values where the function itself equals zero. In other words, they are the points where the parabola representing the quadratic equation intersects the x-axis.
To find these zeros, we use the quadratic formula as previously mentioned. For the equation \(h(x)=3x^2+8x-16\), inserting the coefficients into the formula helps us find these points.

• The zeros obtained from the calculation are \(x = \frac{4}{3}\) and \(x = -4\).

These zeros are part of the solution set and give us the exact points at which the quadratic curve crosses the x-axis. They help in understanding the symmetry and the direction in which the parabola opens, which is either upwards or downwards depending primarily on the sign of the coefficient \(a\).

Step-by-Step Solution

Breaking down the process of finding the zeros of a quadratic function into clear steps can make it more manageable:

• Step 1: Identify Coefficients - From the given quadratic function \(h(x)=3x^2+8x-16\), identify \(a=3\), \(b=8\), and \(c=-16\).
• Step 2: Input Coefficients into the Quadratic Formula - Substitute these coefficients into the quadratic formula: \(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• Step 3: Simplify - Under the square root, perform the arithmetic: \(8^2 - 4 \times 3 \times (-16)\) simplifies to \(256\).
• Step 4: Calculate - Compute the square root, which is \(16\), and determine the potential solutions: \(-8 + 16)/6 = \frac{4}{3}\) and \(-8 - 16)/6 = -4\).

These steps guide you through unraveling the quadratic equation systematically, ensuring that you understand and calculate the zeros accurately.