Problem 45 - Algebra 1: Concepts and Skills

Question

Use the quadratic formula to solve the equation. If the solution involves radicals, round to the nearest hundredth. $$9 x^{2}+14 x+3=0$$

Step-by-Step Solution

Answer

The roots of the given quadratic equation are \(x = -0.26\) and \(x = -1.30\).

Step 1: Identify a, b , and c

In the given equation \(9x^{2} + 14x + 3 = 0\), \(a = 9\), \(b = 14\), and \(c = 3\). These values will be plugged into the quadratic formula.

Step 2: Plug into the quadratic formula

Substitute \(a\), \(b\), and \(c\) into the quadratic formula: \(x = \frac{-14 \pm \sqrt{14^{2} - 4*9*3}}{2*9}\)

Step 3: Simplify inside the square root

First, simplify inside the square root: \(x = \frac{-14 \pm \sqrt{196 - 108}}{18}\). The result inside the square root becomes 88.

Step 4: Simplify the expression

Next, simplify the expression: \(x = \frac{-14 \pm \sqrt{88}}{18}\). The square root of \(88\) is approximately \(9.38\)

Step 5: Complete the operations and round answers

Finally, complete the addition and subtraction operations in the formula and round to the nearest hundredth: \(x = \frac{-14 + 9.38}{18} = -0.26\) and \(x = \frac{-14 - 9.38}{18} = -1.30\)

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a type of polynomial equation where the highest degree of the variable is 2. In general, they can be written in the standard form as:

• \(ax^2 + bx + c = 0\)

Here, \(x\) represents the unknown variable, while \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\). The presence of \(x^2\) makes it a quadratic equation, distinguishing it from linear equations, where the highest power is 1.
The solutions to these equations are referred to as the roots, which can be real or complex numbers. Quadratic equations are widely used in many fields including physics and engineering, as they can represent a variety of real-world scenarios such as projectile motion.

Steps for Solving Quadratics Using the Quadratic Formula

The quadratic formula is a universal method for finding the solutions of quadratic equations. It is expressed as:

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

This formula helps us find the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). Here's how to solve using this formula: Identify \(a\), \(b\), and \(c\): Begin by identifying the coefficients from your quadratic equation. For example, as shown in the step-by-step solution, for the equation \(9x^2 + 14x + 3 = 0\), \(a = 9\), \(b = 14\), and \(c = 3\). Substitute into the formula: Next, substitute these values into the quadratic formula to calculate the roots. Calculate the discriminant: The term under the square root, \(b^2 - 4ac\), is called the discriminant. It provides crucial information about the nature of the roots. A positive discriminant indicates two real and distinct solutions, zero means exactly one real solution, and a negative discriminant suggests two complex solutions.Simplify and solve: Once the discriminant is calculated, the formula will result in two possible solutions by considering the \(\pm\) sign. Complete the arithmetic operations to find these solutions.

Understanding Radicals in the Quadratic Formula

Radicals, often represented by the square root symbol, play a crucial role in solving quadratic equations, particularly when using the quadratic formula. They occur in the expression \( \sqrt{b^2 - 4ac} \) in the quadratic formula.
Radicals are essential for indicating whether the roots of the quadratic equation are rational or irrational, and whether the solutions are real or complex.

• If the expression under the radical (the discriminant) is a perfect square, \( sqrt{b^2 - 4ac}\) yields an integer, leading to rational roots.
• However, if it is not a perfect square, the roots are typically irrational, which might require approximation when expressed numerically.
• If the discriminant is negative, the result involves imaginary numbers, as the square root of a negative number is not real, leading to complex roots.

In this particular problem, the discriminant was 88, which is not a perfect square, resulting in an irrational number (about 9.38) after taking the square root.
Handling radicals accurately is important because they directly affect the type and accuracy of the solutions obtained from quadratic equations.