Resolve the Quadratic Equation: x² + 2x - 24 = -13 + (x-6)(x+6) + 7x

Q: Why does the x² cancel out on both sides?

When you have $ x^2 + 2x - 24 = x^2 + 7x - 49 $, subtracting $ x^2 $ from both sides eliminates the quadratic term. This happens because both sides have the same quadratic term, making it actually a linear equation in disguise!

Q: How do I remember the difference of squares pattern?

Think of it as "First squared minus Last squared": $ (a-b)(a+b) = a^2 - b^2 $. The middle terms cancel out! For $ (x-6)(x+6) $, you get $ x^2 - 6^2 = x^2 - 36 $.

Q: What if I expanded (x-6)(x+6) using FOIL instead?

FOIL works too! You'd get: $ x^2 + 6x - 6x - 36 = x^2 - 36 $. The middle terms cancel, giving the same result as the difference of squares shortcut.

Q: Why isn't this a quadratic equation with two solutions?

Great observation! Even though it starts with $ x^2 $ terms, they cancel out during simplification. The equation becomes linear: $ -5x = -25 $, so there's only one solution.

Q: How can I check if x = 5 is really correct?

Substitute $ x = 5 $ into the original equation:Left side: $ 5^2 + 2(5) - 24 = 25 + 10 - 24 = 11 $Right side: $ -13 + (5-6)(5+6) + 7(5) = -13 + (-1)(11) + 35 = 11 $Both sides equal 11, so $ x = 5 $ is correct!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve a problem!

00:17 We're going to use the quadratic formula.

00:33 First, we'll expand the brackets. Take your time.

00:50 Next, calculate six squared. What's sixty-six?

01:03 Now, let's collect similar terms.

01:10 Arrange the equation so the right side equals zero.

01:27 Again, collect those like terms.

01:41 Let's isolate X.

01:57 And there you have it. That's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Resolve:

$x^2+2x-24=-13+(x-6)(x+6)+7x$

2

Step-by-step solution

To solve the problem, we'll start by simplifying the right side of the equation.

1. Begin by expanding the difference of squares on the right side:
$(x-6)(x+6) = x^2 - 6^2 = x^2 - 36$ .

2. Substitute back into the equation:
$x^2 + 2x - 24 = -13 + (x^2 - 36) + 7x$ .

3. Simplify the right side:
Combine like terms:
$-13 + x^2 - 36 + 7x = x^2 + 7x - 49$ .

4. Now the equation is:
$x^2 + 2x - 24 = x^2 + 7x - 49$ .

5. Subtract $x^2$ from both sides to eliminate $x^2$ :
$2x - 24 = 7x - 49$ .

6. Move all terms involving $x$ to one side and constants to the other side:
Subtract $7x$ from both sides:
$2x - 7x = -49 + 24$ .
Simplify to:
$-5x = -25$ .

7. Solve for $x$ by dividing both sides by $-5$ :
$x = \frac{-25}{-5} = 5$ .

Therefore, the solution to the problem is $x = 5$ .

3

Final Answer

$x=5$

Key Points to Remember

Essential concepts to master this topic

Expansion: Recognize and expand $(x-6)(x+6) = x^2 - 36$ using difference of squares
Simplification: Combine like terms: $-13 + x^2 - 36 + 7x = x^2 + 7x - 49$
Verification: Substitute x = 5 back into original equation to confirm both sides equal ✓

Common Mistakes

Avoid these frequent errors

Incorrectly expanding the difference of squares
Don't expand $(x-6)(x+6)$ as $x^2 + 12x + 36$ = wrong pattern! This treats it like $(x+6)^2$ instead of difference of squares. Always remember $(a-b)(a+b) = a^2 - b^2$ , so $(x-6)(x+6) = x^2 - 36$ .

Practice Quiz

Test your knowledge with interactive questions

Solve:

$$ (2+x)(2-x)=0 $$

FAQ

Everything you need to know about this question

Why does the x² cancel out on both sides?

+

When you have $x^2 + 2x - 24 = x^2 + 7x - 49$ , subtracting $x^2$ from both sides eliminates the quadratic term. This happens because both sides have the same quadratic term, making it actually a linear equation in disguise!

How do I remember the difference of squares pattern?

+

Think of it as "First squared minus Last squared": $(a-b)(a+b) = a^2 - b^2$ . The middle terms cancel out! For $(x-6)(x+6)$ , you get $x^2 - 6^2 = x^2 - 36$ .

What if I expanded (x-6)(x+6) using FOIL instead?

+

FOIL works too! You'd get: $x^2 + 6x - 6x - 36 = x^2 - 36$ . The middle terms cancel, giving the same result as the difference of squares shortcut.

Why isn't this a quadratic equation with two solutions?

+

Great observation! Even though it starts with $x^2$ terms, they cancel out during simplification. The equation becomes linear: $-5x = -25$ , so there's only one solution.

How can I check if x = 5 is really correct?

+

Substitute $x = 5$ into the original equation:

Left side: $5^2 + 2(5) - 24 = 25 + 10 - 24 = 11$
Right side: $-13 + (5-6)(5+6) + 7(5) = -13 + (-1)(11) + 35 = 11$

Both sides equal 11, so $x = 5$ is correct!

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Resolve the Quadratic Equation: x² + 2x - 24 = -13 + (x-6)(x+6) + 7x

Linear Equations with Difference of Squares

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