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Chapter 8: Problem 75

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ x^{2}+8 x+6=0 $$

Short Answer

Expert verified
The solutions are approximately \( x = -0.84 \) and \( x = -7.16 \).

Step by step solution

01

Identify the Quadratic Equation Form

The equation given is a quadratic equation in the standard form: \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = 8 \), and \( c = 6 \).
02

Apply the Quadratic Formula

To find the roots of the equation \( x^2 + 8x + 6 = 0 \), we'll use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
03

Calculate the Discriminant

Compute the discriminant \( b^2 - 4ac \). Substitute \( b = 8 \), \( a = 1 \), \( c = 6 \) into the formula to get: \( 8^2 - 4 \cdot 1 \cdot 6 = 64 - 24 = 40 \).
04

Find the Square Root of the Discriminant

Calculate \( \sqrt{40} \). We approximate \( \sqrt{40} \approx 6.32 \).
05

Compute the Roots Using the Quadratic Formula

Substitute the values into the quadratic formula: \[ x = \frac{-8 \pm 6.32}{2 \times 1} \] This gives two solutions: \[ x_1 = \frac{-8 + 6.32}{2} = \frac{-1.68}{2} \approx -0.84 \] \[ x_2 = \frac{-8 - 6.32}{2} = \frac{-14.32}{2} \approx -7.16 \].
06

Approximate the Solutions

Round the results to the nearest hundredth: The solutions are approximately \( x \approx -0.84 \) and \( x \approx -7.16 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discriminant
The discriminant is a vital part of understanding quadratic equations.
It helps us determine the nature of the roots without actually solving the equation.
For any quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant is expressed as \( b^2 - 4ac \).
Here鈥檚 why it鈥檚 important:
  • If the discriminant is positive, like the 40 in our example, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, it means the equation has exactly one real root (or two identical real roots).
  • If the discriminant is negative, the roots are not real鈥攖hey are complex or imaginary.
In our exercise, calculating the discriminant \( 8^2 - 4 \times 1 \times 6 = 40 \) shows us the equation has two real and distinct solutions.
So we expect to find two valuable roots.
Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation.
Whenever you have an equation in the form \( ax^2 + bx + c = 0 \), this formula provides a quick solution.
The formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula incorporates the discriminant under the square root.
It directly shows us how the value of the discriminant affects the solutions:
  • "\( -b \pm \sqrt{b^2 - 4ac} \)" explains why we often get two answers: one for the plus and one for the minus sign.
  • The denominator "\( 2a \)" shows that our roots are equally split across the x-axis by the factor of the leading coefficient.
Using this formula simplifies the process of solving the equation to find precise solutions.
In our case, substituting \( a = 1 \), \( b = 8 \), and discriminant \( \sqrt{40} \) into this formula was key to solving for \( x \).
Roots of a Quadratic Equation
The roots of a quadratic equation, also known as "solutions" or "zeroes," are the values of \( x \) that make the equation equal zero.
Graphically, they represent where the parabola intersects the x-axis.
Understanding how to find these roots is crucial because it tells us a lot about the behavior of the equation in real-world applications.Here's how we found the roots for our equation:
  • With the quadratic formula: \( x = \frac{-8 \pm 6.32}{2} \), we calculated the two potential solutions by applying the plus and minus signs.
  • The solutions were approximately \( x \approx -0.84 \) and \( x \approx -7.16 \).
  • This effectively means the parabola crosses the x-axis at these points.
These roots help us understand the quadratic's impact and create a complete image of the equation's graphical representation.
Whether used in physics or finance, knowing the roots provides insights into maximum, minimum, and zero-crossing points.

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Most popular questions from this chapter

The vertex of a quadratic function \(f(x)=a x^{2}+b x+c\) is given by the formula \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right) .\) Explain what is meant by the notation \(f\left(-\frac{b}{2 a}\right)\)

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ x^{2}=-\frac{5}{4} x+\frac{3}{2} $$

Describe how to predict what type of solutions the equation \(3 x^{2}-4 x+5=0\) will have.

Determine the coordinates of the vertex of the graph of each function using the vertex formula. Then determine the \(x\) - and \(y\) -intercepts of the graph. Finally, plot several points and complete the graph. See Example \(9 .\) $$ f(x)=-2 x^{2}+8 x-10 $$

Investment Rates. A woman invests $$ 1,000\( in a fund for which interest is compounded annually at a rate \)r .\( After one year, she deposits an additional \)2,000\(. After two years, the balance in the account is $$ 1,000(1+r)^{2}+\$ 2,000(1+r) .\) If this amount is $$ 3,368.10,\( find \)r$
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