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Questions and Answers
What is the key step in solving a linear equation with one variable?
What does the discriminant (b² - 4ac) in the quadratic formula represent?
If the discriminant (b² - 4ac) in the quadratic formula is negative, what type of solutions does the quadratic equation have?
Which of the following steps is typically required when solving equations involving square roots (radicals)?
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In the equation $\sqrt{x + 1} = 3 - (x/5)$, what is the first step to solve for x?
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Which of the following equations is a linear equation with one variable?
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Study Notes
When solving for variables in a mathematical formula, it's essential to understand the different types of formulas and how they behave. Here is some information on these aspects:
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Linear equations: These are simple equations with one variable. To solve them, you can isolate the variable by moving all other terms to one side of the equation. For example, if we have the equation
x + y = 2, we can rearrange it to gety = 2 - x. -
Quadratic equations: These involve squared terms and require more steps to solve. One approach is using the quadratic formula:
x = (-b ± sqrt(b² - 4ac)) / 2aThis expression gives two possible values for
x. Which value is correct depends on whether the discriminantb² - 4acis positive, negative, or zero. If it's positive, both solutions are real numbers; if it's zero, there is only one solution; and if it's negative, the solutions are complex numbers. -
Equations involving radicals (square roots): In such cases, you may need additional steps like clearing fractions, simplifying expressions, and eliminating square roots from the variable term. For instance, consider the equation
√(x + 1) = 3 - (x/5). To solve this, first eliminate the fraction(x/5)by multiplying both sides by5, which results in5√(x + 1) = 15 - 5x. Then, square both sides to remove the square root notation, resulting in25(x + 1) = 225 - 25x. Finally, subtract25xfrom both sides to get25x + 25 = 225 - 25x, which simplifies to50 = 0, indicating no solution exists.
In conclusion, solving for variables involves understanding different types of equations, applying specific methods to isolate the variable, and considering whether the solution will result in multiple or single values.
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Description
Learn about solving equations with one variable, including linear equations, quadratic equations, and equations involving radicals. Understand methods such as isolating the variable, using the quadratic formula, and eliminating square roots to find solutions.
