Solve the Equation: (x-4)² - x(x+8) = 16 Step by Step

Q: Why does this quadratic equation become linear?

Great observation! When we expand and simplify, the $ x^2 $ terms cancel out ($ x^2 - x^2 = 0 $), leaving us with only linear terms. This happens when the coefficient of $ x^2 $ on both sides are equal.

Q: How do I expand $ (x-4)^2 $ correctly?

Use the perfect square formula: $ (a-b)^2 = a^2 - 2ab + b^2 $. So $ (x-4)^2 = x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16 $. Don't forget the middle term!

Q: What if I get confused with the negative signs?

Write each step clearly! For $ -x(x+8) $, distribute the negative to both terms: $ -x \cdot x + (-x) \cdot 8 = -x^2 - 8x $. Take your time with signs - they're crucial!

Q: Should I always expect the quadratic terms to cancel?

No! This problem is special because both sides have the same $ x^2 $ coefficient. Most quadratic equations keep their $ x^2 $ terms and need factoring or the quadratic formula to solve.

Q: How can I check my work before solving?

After expanding, collect like terms carefully. You should have: $ x^2 - 8x + 16 - x^2 - 8x = 16 $. Notice the $ x^2 $ terms will cancel, leaving $ -16x + 16 = 16 $.

Quadratic Equations with Algebraic Simplification

$(x-4)^2-x(x+8)=16$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:04 Use the shortened multiplication formulas to open the parentheses

00:10 Open parentheses properly, multiply by each factor

00:19 Collect like terms

00:34 Isolate X

00:38 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

$(x-4)^2-x(x+8)=16$

Step-by-step solution

To solve this problem, we'll perform the following steps:

Step 1: Expand $(x-4)^2$
Step 2: Expand and simplify $-x(x+8)$
Step 3: Combine like terms to form a quadratic equation
Step 4: Solve the quadratic equation by factoring

Let's go through these steps with detailed explanations:

Step 1: Expand $(x-4)^2$ .

Using the formula $(a-b)^2 = a^2 - 2ab + b^2$ , we get:

$(x-4)^2 = x^2 - 8x + 16$ .

Step 2: Expand $-x(x+8)$ .

Using the distributive property, $-x(x+8) = -x^2 - 8x$ .

Step 3: Combine both parts and simplify:

We have:

$x^2 - 8x + 16 - x^2 - 8x = 16$ .

Simplify by combining like terms:

-16x + 16 = 16.

Subtract 16 from both sides:

-16x = 0.

Solve for $x$ :

-16x = 0 \implies x = 0.

Thus, the solution to the equation is $x = 0$ .

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Expansion: Apply $(a-b)^2 = a^2 - 2ab + b^2$ and distributive property
Technique: Combine like terms: $x^2 - x^2 = 0$ leaves linear equation
Check: Substitute $x = 0$ : $16 - 0 = 16$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute negative signs correctly
Don't expand $-x(x+8)$ as $-x^2 + 8x$ = wrong signs! This gives $-24x = 0$ instead of $-16x = 0$ . Always distribute the negative to both terms: $-x^2 - 8x$ .

Practice Quiz

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$$ (4b-3)(4b-3) $$

Rewrite the above expression as an exponential summation expression:

FAQ

Everything you need to know about this question

Why does this quadratic equation become linear?

Great observation! When we expand and simplify, the $x^2$ terms cancel out ( $x^2 - x^2 = 0$ ), leaving us with only linear terms. This happens when the coefficient of $x^2$ on both sides are equal.

How do I expand $(x-4)^2$ correctly?

Use the perfect square formula: $(a-b)^2 = a^2 - 2ab + b^2$ . So $(x-4)^2 = x^2 - 2(x)(4) + 4^2 = x^2 - 8x + 16$ . Don't forget the middle term!

What if I get confused with the negative signs?

Write each step clearly! For $-x(x+8)$ , distribute the negative to both terms: $-x \cdot x + (-x) \cdot 8 = -x^2 - 8x$ . Take your time with signs - they're crucial!

Should I always expect the quadratic terms to cancel?

No! This problem is special because both sides have the same $x^2$ coefficient. Most quadratic equations keep their $x^2$ terms and need factoring or the quadratic formula to solve.

How can I check my work before solving?

After expanding, collect like terms carefully. You should have: $x^2 - 8x + 16 - x^2 - 8x = 16$ . Notice the $x^2$ terms will cancel, leaving $-16x + 16 = 16$ .

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Solve the Equation: (x-4)² - x(x+8) = 16 Step by Step

Quadratic Equations with Algebraic Simplification

❤️ 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 does this quadratic equation become linear?

How do I expand $(x-4)^2$ correctly?

What if I get confused with the negative signs?

Should I always expect the quadratic terms to cancel?

How can I check my work before solving?

🌟 Unlock Your Math Potential

More Questions

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Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product

Solve the Equation: (x-4)² - x(x+8) = 16 Step by Step

Quadratic Equations with Algebraic Simplification

❤️ 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 does this quadratic equation become linear?

How do I expand (x−4)2 (x-4)^2 (x−4)2 correctly?

What if I get confused with the negative signs?

Should I always expect the quadratic terms to cancel?

How can I check my work before solving?

🌟 Unlock Your Math Potential

More Questions

Square of Difference

Solve (x-1)² - (x+2)² = 15: Difference of Squared Binomials

Solve the Equation: x²+(x-2)²=2(x+1)² with Multiple Squared Terms

Solve the Quadratic Equation: (x-4)² - x(x+8) = 0

Solve (x-4)² = (x+2)(x-1): Perfect Square Equals Product

How do I expand $(x-4)^2$ correctly?