Solve the Quadratic Equation: Find x in 4x² + 2x + 8 = x + 12 + x

Q: Why do I need to simplify both sides first?

Simplifying first makes the equation cleaner and easier to work with! In this problem, combining x + x = 2x on the right side helps you see the pattern more clearly.

Q: Can I solve this without moving everything to one side?

While possible, it's much easier to move all terms to one side to get standard form. This way you can clearly see what type of equation you're dealing with.

Q: How do I know when to use factoring vs. square roots?

When your quadratic simplifies to $ x^2 = \text{number} $, use square roots directly! If you have $ ax^2 + bx + c = 0 $ with a middle term, try factoring first.

Q: Why are there two solutions?

Quadratic equations typically have two solutions because squaring eliminates the sign. When $ x^2 = 1 $, both x = 1 and x = -1 work since $ 1^2 = (-1)^2 = 1 $.

Quadratic Equations with Algebraic Simplification

Solve the following equation:

^{$4x^2+8+2x=x+12+x$}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:08 Let's start by finding the value of X.

00:12 Now, gather all like terms together. This helps us to simplify.

00:21 Next, simplify everything as much as possible.

00:30 Let's isolate X on one side of the equation.

00:48 Great! Now, we need to extract the root.

00:52 Remember, when we extract a root, there are always two solutions: one positive and one negative.

00:59 And that gives us the solution to our problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

^{$4x^2+8+2x=x+12+x$}

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify both sides of the equation.
Step 2: Rearrange terms to form a quadratic equation.
Step 3: Solve the quadratic equation using an appropriate method, such as factoring or applying the quadratic formula.

Now, let's work through each step:

Step 1: Simplify both sides of the equation:

4x^2 + 8 + 2x = x + 12 + x

The right-hand side can be simplified:

x + x + 12 = 2x + 12

This yields the equation:

4x^2 + 8 + 2x = 2x + 12

Step 2: Rearrange the terms to form a quadratic equation. Subtract $2x + 12$ from both sides:

4x^2 + 8 + 2x - 2x - 12 = 0

Combining like terms gives:

4x^2 - 4 = 0

Step 3: Solve the resulting quadratic equation:

First, we add 4 to both sides:

4x^2 = 4

Next, divide both sides by 4:

x^2 = 1

Now, apply the square root to both sides:

x = \pm 1

Therefore, the solutions to the quadratic equation are

$x = \pm 1$ .

The correct answer is choice 2: ±1.

Final Answer

±1

Key Points to Remember

Essential concepts to master this topic

Simplification: Combine like terms on both sides before rearranging
Technique: Move all terms to one side: $4x^2 - 4 = 0$
Check: Substitute x = 1: $4(1)^2 + 2(1) + 8 = 2(1) + 12$ gives 14 = 14 ✓

Common Mistakes

Avoid these frequent errors

Not combining like terms before rearranging
Don't leave x + x as separate terms on the right side = messy equation with extra steps! This makes the problem unnecessarily complicated and increases chance of errors. Always combine like terms first: x + x = 2x.

Practice Quiz

Test your knowledge with interactive questions

Solve the following equation:

$$ 2x^2-8=x^2+4 $$

FAQ

Everything you need to know about this question

Why do I need to simplify both sides first?

Simplifying first makes the equation cleaner and easier to work with! In this problem, combining x + x = 2x on the right side helps you see the pattern more clearly.

Can I solve this without moving everything to one side?

While possible, it's much easier to move all terms to one side to get standard form. This way you can clearly see what type of equation you're dealing with.

How do I know when to use factoring vs. square roots?

When your quadratic simplifies to $x^2 = \text{number}$ , use square roots directly! If you have $ax^2 + bx + c = 0$ with a middle term, try factoring first.

Why are there two solutions?

Quadratic equations typically have two solutions because squaring eliminates the sign. When $x^2 = 1$ , both x = 1 and x = -1 work since $1^2 = (-1)^2 = 1$ .

What if I get a different quadratic form?

No problem! Some equations might not simplify as nicely. If you can't factor easily, use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ .

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