Solve each quadratic equation. If roots are not real, use ii.(3z−1)2+5=0\left(3z-1\right)^2+5=0

z=1±5i3z=\frac{1\pm\sqrt5i}{3}

Solve each quadratic equation. If roots are not real, use i i . ( 3 z − 1 ) 2 + 5 = 0 \left(3z-1\right)^2+5=0

Here, we're going to solve this quadratic equation using our square root property. So we want to isolate our variable z here in order to find what value of z makes this equation true. So to start we're going to subtract five from both sides so that we get three z minus one all squared equals negative five. Now that we have our quantity squared equal to some value, we can apply our square root property. So we're going to square root both sides. So our square root and our square cancel out, and we're left with three z minus one is equal to positive and negative square root of negative five. Next, we'll add one to both sides of the equation. So we have a three z is equal to one plus or minus the square root of negative five. And next, let's divide both sides by three. So that we get z is equal to one plus or minus the square root of negative five all over three. So now we want to simplify our radical. And I know through the product property that I can think of this as the square root of five times the square root of negative one. And we know that the square root of negative one is I and that the square root of five is prime, so it can't be simplified any further. So that means our final answer is z is equal to one plus or minus the square root of five times I all over three. And if we wanted to write these as two separate values, we could also state our answer as z is equal to one plus the square root of five times I all over three. And z is equal to one minuteus the square root of five times I all over three. Both of these would be acceptable answers for our solution set.

Solve each quadratic equation. If roots are not real, use ii.(3z−1)2+5=0\left(3z-1\$$right)^2+5=0$$

z=1±5i3z=\frac{1\pm\sqrt5i}{3}

Indice

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1. Review of Real Numbers2h 39m

Simplifying Fractions
30m

Evaluating Exponents
29m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Translating Sentences to Equations
9m

The Distributive Property
17m

Simplifying Expressions
40m

2. Linear Equations and Inequalities3h 38m

The Addition and Subtraction Properties of Equality
41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
1h 14m

Linear Inequalities in One Variable
1h 11m

3. Solving Word Problems2h 43m

Introduction to Problem Solving
37m

Formulas
21m

Percent Problem Solving
59m

Mixture Problem Solving
43m

4. Graphing Linear Equations in Two Variables3h 17m

The Rectangular Coordinate System
44m

Graph Linear Equations in Two Variables
24m

Graph Linear Equations Using Intercepts
23m

Slope of a Line
44m

Slope-Intercept Form
38m

Point Slope Form
22m

5. Systems of Linear Equations1h 43m

Solving Systems of Linear Equations by Graphing
41m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
45m

6. Exponents and Polynomials3h 25m

The Product Rule
10m

The Quotient Rule
13m

Intro to the Power Rules
18m

The Power of a Quotient Rule
18m

Negative Exponents
28m

Simplifying Exponential Expressions Using All Exponent Rules
6m

Intro to Polynomials
21m

Adding and Subtracting Polynomials
20m

Multiplying Polynomials
33m

Special Products
34m

7. Factoring2h 42m

Factoring by Greatest Common Factor & Grouping
38m

Factoring Trinomials of the Form x² + bx + c
11m

Factoring Trinomials of the Form ax² + bx + c
37m

Factoring Special Products
33m

Quadratic Equations & Applications
41m

8. Rational Expressions and Equations3h 13m

Simplifying Rational Expressions
39m

Multiplying and Dividing Rational Expressions
25m

Adding and Subtracting Rational Expressions with Common Denominators
19m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

9. Inequalities and Absolute Value2h 52m

Set Operations and Compound Inequalities
42m

Absolute Value Equations
38m

Absolute Value Inequalities
39m

Linear Inequalities in Two Variables
28m

Systems of Linear Inequalities
23m

10. Relations and Functions2h 9m

Introduction to Relations and Functions
57m

Function Notation
15m

Graphing Common Functions
5m

Composition of Functions
24m

Direct & Inverse Variation
27m

11. Roots, Radicals, and Complex Numbers3h 57m

Introduction to Radical Expressions
47m

Simplifying Radical Expressions
47m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
53m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

12. Quadratic Equations and Functions3h 1m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
40m

Graphing Quadratic Equations
1h 28m

13. Inverse, Exponential, & Logarithmic Functions2h 31m

Introduction to Inverse Functions
25m

Exponential Functions
35m

Introduction to Logarithmic Functions
33m

Properties of Logarithms
55m

14. Conic Sections & Systems of Nonlinear Equations2h 24m

Graphing Circles
32m

Ellipses
35m

Parabolas
35m

Hyperbolas
41m

15. Sequences, Series, and the Binomial Theorem1h 45m

Introduction to Sequences
40m

Arithmetic Sequences
27m

Geometric Sequences
26m

Factorials
11m

12. Quadratic Equations and Functions

The Square Root Property

Beginning & Intermediate Algebra

12. Quadratic Equations and Functions

The Square Root Property

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Solve each quadratic equation. If roots are not real, use $i$ .
${(3 z - 1)}^{2} + 5 = 0$

$z = \frac{1 \pm 2 i}{3}$

$z = \frac{1 \pm \sqrt{5} i}{3}$

$z = \pm 3 i$

$z = \pm \frac{1}{3} i$

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Problema precedenteProssimo problema

Guida verificata passo dopo passo

Start with the given equation: $\left(3z - 1\right)^2 + 5 = 0$.

Isolate the squared term by subtracting 5 from both sides: $\left(3z - 1\right)^2 = -5$.

Take the square root of both sides, remembering to include the plus-minus symbol and the imaginary unit $i$ because the right side is negative: $3z - 1 = \pm \sqrt{-5} = \pm \sqrt{5}i$.

Solve for $z$ by adding 1 to both sides: $3z = 1 \pm \sqrt{5}i$.

Finally, divide both sides by 3 to isolate $z$: $z = \frac{1 \pm \sqrt{5}i}{3}$.