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Magazine Chapter 1: Problem 85
Find the real solutions, if any, of each equation. Use any method. $$ 2+z=6 z^{2} $$
Short Answer
Expert verified
The real solutions are \(z = \frac{2}{3}\) and \(z = -\frac{1}{2}\).
Step by step solution
01
Rewrite the Equation
First, rewrite the given equation in standard quadratic form: \[2 + z = 6z^2\]
02
Move All Terms to One Side
Subtract 2 and z from both sides to set the equation to zero: \[0 = 6z^2 - z - 2\] or equivalently, \[6z^2 - z - 2 = 0\]
03
Identify Coefficients
Identify the coefficients in the quadratic equation: \[a = 6,\; b = -1,\; c = -2\]
04
Apply the Quadratic Formula
Apply the quadratic formula \[z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] with the identified coefficients: \[z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4\cdot6\cdot(-2)}}{2\cdot6}\]
05
Simplify the Radicand
Simplify the expression under the square root (the radicand): \[(-1)^2 - 4\cdot6\cdot(-2) = 1 + 48 = 49\]
06
Calculate the Square Root
Evaluate the square root of 49: \[\sqrt{49} = 7\]
07
Solve for z
Substitute the simplified values back into the quadratic formula and solve for \(z\): \[z = \frac{1 \pm 7}{12}\] This gives two solutions: \[z = \frac{1 + 7}{12} = \frac{8}{12} = \frac{2}{3}\] and \[z = \frac{1 - 7}{12} = \frac{-6}{12} = -\frac{1}{2}\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Solving Quadratic Equations
To solve a quadratic equation, you need to find the values of the variable that make the equation true. Quadratic equations typically have the form \[ax^2 + bx + c = 0\] where \(a, b,\) and \(c\) are constants. You can use several methods to solve these equations, including:
- Factoring
- Completing the square
- Using the Quadratic Formula
- Graphing
Quadratic Formula
The Quadratic Formula provides a straightforward way to find solutions to any quadratic equation, regardless of whether it can be factored. The formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] Here鈥檚 how it works:
- Identify the coefficients \(a, b,\) and \(c\).
- Substitute these coefficients into the formula.
- Calculate the discriminant, \(b^2 - 4ac\).
- Solve the resulting expression to find the values of \(x\).
Identifying Coefficients
Identifying the coefficients in a quadratic equation is a crucial step. These constants \(a, b,\) and \(c\) will be substituted into the Quadratic Formula or used in other solution methods. For the equation \(6z^2 - z - 2 = 0\), the coefficients were:
- \(a = 6\): The coefficient of \(z^2\)
- \(b = -1\): The coefficient of \(z\)
- \(c = -2\): The constant term
Simplifying Radicals
Simplifying radicals is an essential skill when solving quadratic equations using the Quadratic Formula. In the context of our exercise, the discriminant \(b^2 - 4ac\) turned out to be 49. Simplifying this radical involved finding \(\sqrt{49} = 7\). Here are some tips to simplify radicals:
- Look for perfect square factors.
- Break down the number under the radical into its prime factors.
- Simplify the expression by taking the square root of any perfect squares.
