Problem 69 Solve each equation. Give both t... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 69

Solve each equation. Give both the exact answer and a decimal approximation to the nearest tenth. $$x^{2}+1=-4 x$$

Short Answer

Expert verified

$x = -2 + \sqrt{3}, -2 - \sqrt{3}$ (exact) and $x \approx -0.3, -3.7$ (approximate to the nearest tenth)

Step by step solution

Rewrite the given equation in the standard form

The standard form of a quadratic equation is $ ax^2 + bx + c = 0 $. Our equation can be rewritten as $ x^2 + 4x + 1 = 0 $ by adding $4x$ and subtracting $1$ from both sides.

Identify the values of $a$, $b$, and $c$

The coefficient of $x^2$ is $a$, the coefficient of $x$ is $b$, and the constant term is $c$. Comparing $x^2 + 4x + 1 = 0$ with the general form, we find $a = 1$, $b = 4$, and $c = 1$.

Substitute $a$, $b$, and $c$ into the quadratic formula

Substituting these values into the quadratic formula gives us $x = \frac{-4\pm\sqrt{(4)^2 - 4(1)(1)}}{2*1}$. This simplifies to $x = \frac{-4\pm\sqrt{16 - 4}}{2}$.

Calculate the two possible values for $x$

After simplifying, we find that the two possible values for $x$ are: $x = \frac{-4+\sqrt{12}}{2}$ and $x = \frac{-4-\sqrt{12}}{2}$. We simplify these more to find that the exact answers are $x = -2 + \sqrt{3}$ and $x = -2 - \sqrt{3}$.

Convert the exact answers to decimal approximations

The value of $\sqrt{3}$ is approximately 1.7320508075689. So, the decimal approximations of the two exact values are: $x_1 = -2 + 1.7320508075689 \approx -0.3$, and $x_2 = -2 - 1.7320508075689 \approx -3.7$. But, as we need to round up to the nearest tenth. The final decimal approximations are $x_1 \approx -0.3$ and $x_2 \approx -3.7$.

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Key Concepts

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

Standard Form

Quadratic equations are often easier to solve when they are in a neat and orderly arrangement known as the standard form. If you've ever worked with quadratic equations, you've likely encountered something like this: \[ ax^2 + bx + c = 0 \] Think of the standard form as a tidy way to see all the parts of a quadratic equation in one place. Here, 'a', 'b', and 'c' are constants where:

'a' is the coefficient of $ x^2 $, which shouldn't be zero,
'b' is the coefficient of $ x $, and
'c' is the constant term.

Getting an equation into the standard form often involves moving terms around. For example, the exercise equation $ x^2 + 1 = -4x $ becomes $ x^2 + 4x + 1 = 0 $ after rearranging it.By doing this, we set the stage to apply methods like the quadratic formula for solving these equations.

Quadratic Formula

Once an equation is in the standard form, the quadratic formula comes into play. This formula is a tool that provides the solutions for quadratic equations. The quadratic formula looks like this: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The "$ \pm $" sign means there are often two solutions. These represent different possible solutions for 'x'. In our example, plugging $ a = 1 $, $ b = 4 $, and $ c = 1 $ into the formula gives: \[ x = \frac{-4 \pm \sqrt{4^2 - 4 \times 1 \times 1}}{2 \times 1} \] Simplifying this helps us find the exact values of 'x', using simple mathematics.

Decimal Approximation

Finding exact solutions is crucial, but sometimes having a decimal approximation is just as useful. It makes it easier to grasp the size of a number, especially when you encounter square roots or fractions. When simplifying, you might end up with expressions like $-2 + \sqrt{3}$. To transform these, you use the approximate value of the square root, such as $ \sqrt{3} \approx 1.732 $. For our example:

$ x_1 = -2 + 1.732 \approx -0.3 $
$ x_2 = -2 - 1.732 \approx -3.7 $

Decimal approximations help us to visualize and understand numbers better. In practical scenarios, rounding to the nearest tenth makes it even simpler! This way, mathematical solutions become more relatable and easier to communicate in everyday terms.

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91影视

Solve each equation. Give both the exact answer and a decimal approximation to the nearest tenth. $$x^{2}+1=-4 x$$

Short Answer

Step by step solution

Rewrite the given equation in the standard form

Identify the values of \(a\), \(b\), and \(c\)

Substitute \(a\), \(b\), and \(c\) into the quadratic formula

Calculate the two possible values for \(x\)

Convert the exact answers to decimal approximations

Key Concepts

Standard Form

Quadratic Formula

Decimal Approximation

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