Solve the Quadratic Equation: x² - 13x + 3 + (x-3)(x+3) = (x-6)(x+6)

Q: What's the difference of squares formula?

The difference of squares formula is $ (a-b)(a+b) = a^2 - b^2 $. Notice how the middle terms cancel out! For example: $ (x-3)(x+3) = x^2 - 9 $.

Q: Why don't I use FOIL here?

You can use FOIL, but recognizing the difference of squares pattern is much faster! FOIL gives $ x^2 + 3x - 3x - 9 = x^2 - 9 $, but the pattern lets you skip straight to the answer.

Q: How do I combine like terms correctly?

Group terms with the same variable and power: $ x^2 + x^2 = 2x^2 $Constants: $ 3 + (-9) = -6 $Keep $ -13x $ as is

Q: What if I can't factor the final quadratic?

If factoring doesn't work easily, try the quadratic formula: $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $. But for $ x^2 - 13x + 30 $, look for two numbers that multiply to 30 and add to -13.

Q: How do I check if x = 3 and x = 10 are correct?

Substitute each value into the original equation:For x = 3: Both sides equal -18For x = 10: Both sides equal -33Since both sides match for each value, our solutions are correct!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:06 Let's use the shortened multiplication formulas

00:37 Calculate 3 squared and 6 squared

00:45 Simplify what we can

00:58 Arrange the equation so that the right side equals 0

01:09 Group terms

01:20 Use the shortened multiplication formulas and find two numbers

01:23 whose sum equals minus 13

01:27 and their product equals 30

01:34 These are the numbers

01:45 Find the two possible solutions that make the equation equal zero

01:59 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Resolve:

^{$x^2-13x+3+(x-3)(x+3)=(x-6)(x+6)$}

2

Step-by-step solution

To solve this quadratic equation, follow these steps:

Step 1: Expand both sides completely.
Step 2: Simplify both sides to form a quadratic equation.
Step 3: Solve the quadratic equation using appropriate methods.

Now, let's work through the steps:

Step 1: Expand the expressions.

The left side is $x^2 - 13x + 3 + (x-3)(x+3)$ . Using the difference of squares formula, expand $(x-3)(x+3)$ as:

(x-3)(x+3) = x^2 - 3^2 = x^2 - 9

Substituting back, the left side becomes:

x^2 - 13x + 3 + x^2 - 9

The right side is $(x-6)(x+6)$ . Expand using the difference of squares:

(x-6)(x+6) = x^2 - 6^2 = x^2 - 36

Step 2: Simplify both sides.

Combine terms on the left side:

x^2 - 13x + 3 + x^2 - 9 = 2x^2 - 13x - 6

The right side remains:

x^2 - 36

The equation becomes:

2x^2 - 13x - 6 = x^2 - 36

Subtract $x^2$ from both sides to simplify further:

2x^2 - 13x - 6 - x^2 = -36

x^2 - 13x - 6 = -36

Step 3: Solve for $x$ .

Move -36 to the other side to form a standard quadratic equation:

x^2 - 13x - 6 + 36 = 0

x^2 - 13x + 30 = 0

Factor the quadratic:

(x - 3)(x - 10) = 0

Setting each factor to zero gives:

x - 3 = 0 \quad \text{or} \quad x - 10 = 0

These lead to the solutions:

x = 3 \quad \text{and} \quad x = 10

Thus, the solutions to the equation are $x = 3$ and $x = 10$ .

3

Final Answer

10,3

Key Points to Remember

Essential concepts to master this topic

Pattern Recognition: Use $(a-b)(a+b) = a^2 - b^2$ formula
Simplification: Combine like terms: $2x^2 - 13x - 6 = x^2 - 36$
Verification: Substitute $x = 3$ and $x = 10$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Expanding (x-3)(x+3) incorrectly as x² + 6x + 9
Don't use FOIL on difference of squares = wrong middle term appears! This creates extra x terms that don't belong. Always recognize (x-3)(x+3) = x² - 9 directly using the difference of squares pattern.

Practice Quiz

Test your knowledge with interactive questions

Solve:

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

FAQ

Everything you need to know about this question

What's the difference of squares formula?

+

The difference of squares formula is $(a-b)(a+b) = a^2 - b^2$ . Notice how the middle terms cancel out! For example: $(x-3)(x+3) = x^2 - 9$ .

Why don't I use FOIL here?

+

You can use FOIL, but recognizing the difference of squares pattern is much faster! FOIL gives $x^2 + 3x - 3x - 9 = x^2 - 9$ , but the pattern lets you skip straight to the answer.

How do I combine like terms correctly?

+

Group terms with the same variable and power:

$x^2 + x^2 = 2x^2$
Constants: $3 + (-9) = -6$
Keep $-13x$ as is

What if I can't factor the final quadratic?

+

If factoring doesn't work easily, try the quadratic formula: $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . But for $x^2 - 13x + 30$ , look for two numbers that multiply to 30 and add to -13.

How do I check if x = 3 and x = 10 are correct?

+

Substitute each value into the original equation:

For x = 3: Both sides equal -18
For x = 10: Both sides equal -33

Since both sides match for each value, our solutions are correct!

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Solve the Quadratic Equation: x² - 13x + 3 + (x-3)(x+3) = (x-6)(x+6)

Quadratic Equations with Difference of Squares

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