Resolve the Equation: 5x² + 7x + 9 = (2x-1)(2x+1)

Q: Why does $ (2x-1)(2x+1) $ equal $ 4x^2 - 1 $ ?

This is a difference of squares pattern! When you multiply $ (a-b)(a+b) $, the middle terms cancel out. Here: $ 2x \cdot 2x + 2x \cdot 1 - 1 \cdot 2x - 1 \cdot 1 = 4x^2 + 2x - 2x - 1 = 4x^2 - 1 $

Q: How do I know when to use factoring vs. the quadratic formula?

Try factoring first if the equation looks simple! For $ x^2 + 7x + 10 = 0 $, look for two numbers that multiply to 10 and add to 7: that's 2 and 5, so $ (x+2)(x+5) = 0 $.

Q: Why are both answers negative?

The solutions $ x = -2 $ and $ x = -5 $ are both negative because when we factor $ x^2 + 7x + 10 $, we get $ (x+2)(x+5) = 0 $. Setting each factor to zero gives us negative values.

Q: What if I get different answers using different methods?

Check your algebra! The quadratic formula and factoring should give the same answers. If they don't match, review your expansion of the right side and your arithmetic when collecting like terms.

Q: Do I need to check both solutions?

Yes, always verify both! Substitute each answer back into the original equation. Both $ x = -2 $ and $ x = -5 $ should make both sides equal when plugged in.

Quadratic Equations with Difference of Squares

Solve the following equation:

^{$5x^{2}+7x+9=(2x-1)(2x+1)$}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve

00:11 Let's use the abbreviated multiplication formulas

00:35 Open the parentheses, calculate 2 squared

00:44 Arrange the equation so that the right side equals 0

01:00 Collect terms

01:11 Use the abbreviated multiplication formulas and find two numbers

01:15 whose sum equals 7

01:19 and their product equals 10

01:24 These are the matching numbers

01:31 Find the possible solutions that make the equation equal zero

01:45 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

^{$5x^{2}+7x+9=(2x-1)(2x+1)$}

Step-by-step solution

Let's begin by focusing on the right side of the equation:

$(2x-1)(2x+1)$

We must first open the parentheses whilst multiplying all the terms as needed:

$2x\cdot2x+2x\cdot1+-1\cdot2x+-1\cdot1$

$4x^2+2x-2x-1$
$4x^2-1$

Let's now return to the original equation, and move all terms to the same side.

$5X^2+7x+9=4X^2-1$
$5X^2-4X^2+7x+9+1=0$
$X^2+7X+10=0$

We are left with a simple quadratic equation, which can be solved using any method we desire (factoring or the quadratic formula).

Therefore the final solution is:

$X= -2,-5$

Final Answer

2-,5-

Key Points to Remember

Essential concepts to master this topic

Rule: Recognize $(2x-1)(2x+1) = 4x^2 - 1$ as difference of squares
Technique: Expand right side first: $4x^2 - 1$ , then collect like terms
Check: Substitute x = -2: $5(-2)^2 + 7(-2) + 9 = 15$ and $4(-2)^2 - 1 = 15$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand the right side correctly
Don't just multiply $2x \cdot 2x = 4x^2$ and stop there = missing the middle terms! This gives $4x^2$ instead of $4x^2 - 1$ . Always use FOIL: First, Outer, Inner, Last to get all four products.

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 $(2x-1)(2x+1)$ equal $4x^2 - 1$ ?

This is a difference of squares pattern! When you multiply $(a-b)(a+b)$ , the middle terms cancel out. Here: $2x \cdot 2x + 2x \cdot 1 - 1 \cdot 2x - 1 \cdot 1 = 4x^2 + 2x - 2x - 1 = 4x^2 - 1$

How do I know when to use factoring vs. the quadratic formula?

Try factoring first if the equation looks simple! For $x^2 + 7x + 10 = 0$ , look for two numbers that multiply to 10 and add to 7: that's 2 and 5, so $(x+2)(x+5) = 0$ .

Why are both answers negative?

The solutions $x = -2$ and $x = -5$ are both negative because when we factor $x^2 + 7x + 10$ , we get $(x+2)(x+5) = 0$ . Setting each factor to zero gives us negative values.

What if I get different answers using different methods?

Check your algebra! The quadratic formula and factoring should give the same answers. If they don't match, review your expansion of the right side and your arithmetic when collecting like terms.

Do I need to check both solutions?

Yes, always verify both! Substitute each answer back into the original equation. Both $x = -2$ and $x = -5$ should make both sides equal when plugged in.

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Resolve the Equation: 5x² + 7x + 9 = (2x-1)(2x+1)

Quadratic Equations with Difference of Squares

❤️ 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 $(2x-1)(2x+1)$ equal $4x^2 - 1$ ?

How do I know when to use factoring vs. the quadratic formula?

Why are both answers negative?

What if I get different answers using different methods?

Do I need to check both solutions?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

Solve the Quadratic Equation: (√x + 1/2)(√x - 1/2) = 0

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

Simplify and Solve: Find x in (x-4)(x+4) - 5x = -8 - x² - 25x + 4x - 38

Resolve the Equation: 5x² + 7x + 9 = (2x-1)(2x+1)

Quadratic Equations with Difference of Squares

❤️ 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 (2x−1)(2x+1) (2x-1)(2x+1) (2x−1)(2x+1) equal 4x2−1 4x^2 - 1 4x2−1?

How do I know when to use factoring vs. the quadratic formula?

Why are both answers negative?

What if I get different answers using different methods?

Do I need to check both solutions?

🌟 Unlock Your Math Potential

More Questions

Difference of squares

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

Solve the Quadratic Equation: (√x + 1/2)(√x - 1/2) = 0

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

Simplify and Solve: Find x in (x-4)(x+4) - 5x = -8 - x² - 25x + 4x - 38

Why does $(2x-1)(2x+1)$ equal $4x^2 - 1$ ?