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 ± 5 i 3 z=\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 i i . ( 3 z − 1 ) 2 + 5 = 0 \left(3z-1\right)^2+5=0

z = 1 ± 5 i 3 z=\frac{1\pm\sqrt5i}{3}

z = 1 ± 2 i 3 z=\frac{1\pm2i}{3}

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1. Review of Real Numbers1h 28m

Evaluating Exponents
15m

Evaluating Expressions
15m

Translating Phrases to Expressions
16m

Simplifying Expressions
40m

2. Linear Equations and Inequalities2h 18m

The Addition and Subtraction Properties of Equality
41m

The Multiplication and Division Properties of Equality
30m

Solving Linear Equations
19m

Linear Inequalities in One Variable
46m

3. Solving Word Problems1h 22m

Introduction to Problem Solving
24m

Formulas
21m

Percent Problem Solving
36m

4. Graphing Linear Equations in Two Variables1h 50m

The Rectangular Coordinate System
27m

Graph Linear Equations Using Intercepts
11m

Slope of a Line
33m

Slope-Intercept Form
25m

Point Slope Form
13m

5. Systems of Linear Equations1h 25m

Solving Systems of Linear Equations by Graphing
41m

Solving Systems of Linear Equations by Substitution
15m

Solving Systems of Linear Equations by Elimination
27m

7. Factoring1h 30m

Factoring by Greatest Common Factor & Grouping
26m

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

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

Quadratic Equations & Applications
14m

8. Rational Expressions and Equations2h 18m

Simplifying Rational Expressions
29m

Least Common Denominators
32m

Adding and Subtracting Rational Expressions with Different Denominators
32m

Rational Equations
44m

9. Inequalities and Absolute Value42m

Set Operations and Compound Inequalities
42m

10. Relations and Functions1h 10m

Introduction to Relations and Functions
25m

Function Notation
15m

Graphing Common Functions
5m

Composition of Functions
24m

11. Roots, Radicals, and Complex Numbers2h 33m

Introduction to Radical Expressions
11m

Simplifying Radical Expressions
27m

Adding and Subtracting Radicals
19m

Multiplying, Dividing, and Rationalizing Radicals
23m

Introduction to Complex Numbers
18m

Multiplying and Dividing Complex Numbers
51m

12. Quadratic Equations and Functions1h 23m

The Square Root Property
25m

Completing the Square
26m

The Quadratic Formula
31m

13. Inverse, Exponential, & Logarithmic Functions1h 5m

Introduction to Inverse Functions
25m

Exponential Functions
13m

Introduction to Logarithmic Functions
26m

14. Conic Sections & Systems of Nonlinear Equations58m

Graphing Circles
32m

Ellipses
15m

Parabolas
10m

15. Sequences, Series, and the Binomial Theorem Coming soon

Introduction to Sequences
0m

Arithmetic Sequences
0m

Factorials
0m

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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Multiple Choice

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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Verified step by step guidance

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

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