Solve the Quadratic Equation: Find X in (3x+1)² + 8 = 12

Q: Why do I get two answers for x?

Quadratic equations naturally have two solutions because when you square a number, both positive and negative values give the same result. For example, $ 2^2 = 4 $ and $ (-2)^2 = 4 $.

Q: What does the ± symbol mean exactly?

The ± symbol means "plus or minus" - it's a shorthand for writing two separate equations. $ 3x + 1 = ±2 $ means solve both $ 3x + 1 = 2 $ AND $ 3x + 1 = -2 $.

Q: Do I always isolate the squared term first?

Yes! Always move constants to one side before taking square roots. This makes the equation much simpler: $ (3x+1)^2 = 4 $ is easier to solve than $ (3x+1)^2 + 8 = 12 $.

Q: How do I check if both answers are correct?

Substitute each solution back into the original equation. Both $ x = \frac{1}{3} $ and $ x = -1 $ should make the left side equal 12.

Q: What if the number under the square root is negative?

If you get something like $ (3x+1)^2 = -4 $, there's no real solution because you can't take the square root of a negative number in real numbers.

Q: Can I expand the squared term instead?

You could expand $ (3x+1)^2 $ to $ 9x^2 + 6x + 1 $, but taking the square root directly is much faster and easier!

Quadratic Equations with Perfect Square Forms

Find X

$(3x+1)^2+8=12$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Open brackets properly according to quadratic formula

00:13 Calculate squares and multiplications

00:24 Arrange the equation so the right side equals 0

00:31 Reduce as much as possible

00:37 Examine the coefficients

00:41 Use the root formula to find possible solutions

00:50 Substitute appropriate values and solve for solutions

01:07 Calculate the square and multiplications

01:18 Calculate square root of 16

01:23 These are the 2 options (positive and negative)

01:33 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find X

$(3x+1)^2+8=12$

Step-by-step solution

To solve the equation $(3x + 1)^2 + 8 = 12$ , we start by isolating the squared expression:

First, subtract 8 from both sides to simplify: $(3x + 1)^2 = 12 - 8$ .
This gives $(3x + 1)^2 = 4$ .

Next, we take the square root of both sides to remove the square:

$3x + 1 = \pm 2$ , recognizing the positive and negative roots of 4.

We now solve for $x$ in each case:

Case 1: $3x + 1 = 2$
Subtract 1 from both sides: $3x = 1$
Divide by 3: $x = \frac{1}{3}$ .
Case 2: $3x + 1 = -2$
Subtract 1 from both sides: $3x = -3$
Divide by 3: $x = -1$ .

Therefore, the solutions to the original equation are $x_1 = \frac{1}{3}$ and $x_2 = -1$ .

Final Answer

$x_1=\frac{1}{3},x_2=-1$

Key Points to Remember

Essential concepts to master this topic

Isolation: Move constant terms to isolate the squared expression first
Square Root: Take ±√ of both sides: $\sqrt{4} = ±2$
Verification: Check both solutions in original equation: $(3(\frac{1}{3})+1)^2 + 8 = 12$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the ± when taking square roots
Don't write $3x + 1 = 2$ only = missing one solution! Every positive number has both positive and negative square roots. Always write $3x + 1 = ±2$ to find both solutions.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why do I get two answers for x?

Quadratic equations naturally have two solutions because when you square a number, both positive and negative values give the same result. For example, $2^2 = 4$ and $(-2)^2 = 4$ .

What does the ± symbol mean exactly?

The ± symbol means "plus or minus" - it's a shorthand for writing two separate equations. $3x + 1 = ±2$ means solve both $3x + 1 = 2$ AND $3x + 1 = -2$ .

Do I always isolate the squared term first?

Yes! Always move constants to one side before taking square roots. This makes the equation much simpler: $(3x+1)^2 = 4$ is easier to solve than $(3x+1)^2 + 8 = 12$ .

How do I check if both answers are correct?

Substitute each solution back into the original equation. Both $x = \frac{1}{3}$ and $x = -1$ should make the left side equal 12.

What if the number under the square root is negative?

If you get something like $(3x+1)^2 = -4$ , there's no real solution because you can't take the square root of a negative number in real numbers.

Can I expand the squared term instead?

You could expand $(3x+1)^2$ to $9x^2 + 6x + 1$ , but taking the square root directly is much faster and easier!

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Solve the Quadratic Equation: Find X in (3x+1)² + 8 = 12

Quadratic Equations with Perfect Square Forms

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I get two answers for x?

What does the ± symbol mean exactly?

Do I always isolate the squared term first?

How do I check if both answers are correct?

What if the number under the square root is negative?

Can I expand the squared term instead?

🌟 Unlock Your Math Potential

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