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Chapter 8: Problem 16

Find all complex-number solutions. $$ 7 x^{2}-5=0 $$

Short Answer

Expert verified
x = \pm \frac{\sqrt{35}}{7}

Step by step solution

01

Isolate the quadratic term

Move the constant term to the other side of the equation. This gives: \[ 7x^2 = 5 \]
02

Solve for the quadratic term

Divide both sides of the equation by 7 to isolate the square term:\[ x^2 = \frac{5}{7} \]
03

Find the square root of both sides

Take the square root of both sides of the equation. Remember to include both the positive and negative roots:\[ x = \pm \sqrt{\frac{5}{7}} \]
04

Simplify the expression

Simplify the expression under the square root if possible. Here, we can express it as:\[ x = \pm \frac{\sqrt{5}}{\sqrt{7}} \]
05

Rationalize the denominator

Rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{7}\):\[ x = \pm \frac{\sqrt{5} \times \sqrt{7}}{7} = \pm \frac{\sqrt{35}}{7} \]

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Key Concepts

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

Complex Number Solutions
When you encounter a quadratic equation like \(7 x^{2}-5=0\) and solve it, you might get solutions that involve square roots. Not all solutions to quadratic equations are real numbers.
They can also be complex numbers. Complex numbers include a real part and an imaginary part, like \(a + bi\), where \(i\) is the imaginary unit.
In our example, the solutions do not have an imaginary part, since we get real roots. But in many cases, especially when the square root is of a negative number, it becomes complex.
Understanding complex numbers helps you solve a broader range of equations.
Quadratic Term Isolation
The first step in solving a quadratic equation like \(7 x^{2}-5=0\) is to isolate the quadratic term.
This means getting the term with the variable squared by itself on one side of the equation. For example:
  • Move any constants to the opposite side by adding or subtracting from both sides.
  • Divide every term by the coefficient of the quadratic term to set it to 1.
In our equation, isolating the quadratic term gives us: 7x^2 = 5 This sets up the equation better for the next steps and makes it simpler to solve.
Rationalizing the Denominator
When you have an expression with a square root in the denominator like \(x = \pm \frac{\sqrt{5}}{\sqrt{7}}\), you often want to rationalize it.
This means converting the expression so that the denominator is a whole number. Here鈥檚 how to do it:
  • Multiply both the numerator and the denominator by the same square root to eliminate it from the denominator.
For instance, in our example: \frac{\sqrt{5}}{\sqrt{7}} = \frac{\sqrt{5} \times \sqrt{7}}{7} = \frac{\sqrt{35}}{7} Rationalizing makes the expression cleaner and easier to work with in further calculations.

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Most popular questions from this chapter

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