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Chapter 10: Problem 40

Solve using the Square Root Property. \(\left(q-\frac{3}{5}\right)^{2}=\frac{3}{4}\)

Short Answer

Expert verified
q = \( \frac{3}{5} \pm \frac{\sqrt{3}}{2} \)

Step by step solution

01

Isolate the squared term

The equation is already in the form where the squared term is isolated: ewline \[ \left(q-\frac{3}{5}\right)^2 = \frac{3}{4} \]
02

Apply the square root property

Take the square root of both sides of the equation. Remember to consider both the positive and negative roots: ewline \[ \sqrt{\left(q-\frac{3}{5}\right)^2} = \pm \sqrt{\frac{3}{4}} \]
03

Simplify the equation

Since the square root and the square cancel each other out, we obtain: ewline \[ q-\frac{3}{5} = \pm \frac{\sqrt{3}}{2} \]
04

Solve for q

Isolate q by adding \( \frac{3}{5} \) to both sides of the equation: ewline \[ q = \frac{3}{5} \pm \frac{\sqrt{3}}{2} \]
05

Provide the two solutions

Finally, split the equation into its two possible values for q: ewline \[ q = \frac{3}{5} + \frac{\sqrt{3}}{2} \] ewline \[ q = \frac{3}{5} - \frac{\sqrt{3}}{2} \]

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Key Concepts

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

Solving Equations
Solving equations means finding the value(s) of the variable(s) that make the equation true. When solving equations, you want to isolate the variable on one side of the equation. Here鈥檚 how you can do it:
  • Combine like terms to simplify the expression.
  • Use inverse operations to isolate the variable. For instance, if the variable is added by a number, subtract that number from both sides.
  • Keep the equation balanced by performing the same operation on both sides.
In our example, we started with the equation: \ \ \ \[\left(q-\frac{3}{5}\right)^2=\frac{3}{4}\]
The squared term is already isolated. Next, we took the square root of both sides to further solve for the variable.
Quadratic Equations
A quadratic equation is a second-degree polynomial equation in the form of \[ax^2+bx+c=0\] Qquadratic equations can have one, two, or no real solutions. Common methods to solve them include:
  • Factoring: Expressing the quadratic as a product of binomials.
  • Completing the square: Making a perfect square trinomial.
  • Using the Quadratic Formula: Applying \[x=\frac{{-b\pm\sqrt{{b^2-4ac}}}}{2a}\]
In our example, we used the Square Root Property, which applies when the quadratic is in the form: \[\left(x+c\right)^2=k\]
Taking the square root of both sides gives us: \[x+c=\pm\sqrt{k}\]
Radicals
Radicals are expressions that include square roots, cube roots, or other roots. The square root of a number is represented as: \[\sqrt{x}\]
Here are key points for working with radicals:
  • The principal square root is the non-negative root of a number.
  • Remember to consider both positive and negative roots when solving equations.
  • Simplify radicals by factoring inside the square root.
In solving our equation, we took \[\sqrt{\left(q-\frac{3}{5}\right)^{2}}\] giving us: \[q-\frac{3}{5}=\pm\sqrt{\frac{3}{4}}\] This is crucial in arriving at the solutions: \[q=\frac{3}{5}\pm\frac{\sqrt{3}}{2}\]

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Most popular questions from this chapter

Solve by using the Quadratic Formula. \(\frac{1}{9} c^{2}+\frac{2}{3} c=3\)

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Solve by completing the square. \(2 p^{2}+7 p=14\)

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