Solve the Quadratic: Expanding (x+3)² to Find x in 2x+5

Q: How do I remember the expansion formula $ (a+b)^2 $ ?

Think "First, Outer, Inner, Last" or use the pattern: square the first term, add twice the product of both terms, then square the last term. For $ (x+3)^2 $: $ x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9 $.

Q: What does it mean that $ x^2 + 4x + 4 = (x+2)^2 $ ?

This is called a perfect square trinomial because it factors into a perfect square! When $ (x+2)^2 = 0 $, the only solution is $ x = -2 $ (called a repeated root).

Q: Could I solve this using the quadratic formula instead?

Yes! Using $ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} $ with $ a=1, b=4, c=4 $ gives $ x = \frac{-4 \pm 0}{2} = -2 $. But recognizing the perfect square trinomial is faster!

Q: How can I check if my factoring is correct?

Expand your factored form! If $ (x+2)^2 $ is correct, then $ (x+2)^2 = x^2 + 4x + 4 $, which matches our simplified equation.

Quadratic Equations with Perfect Square Trinomials

Solve the following equation:

$(x+3)^2=2x+5$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the abbreviated multiplication formulas

00:13 Substitute appropriate values according to the data and open the parentheses

00:30 Substitute in our equation

00:50 Arrange the equation so that one side equals 0

00:58 Collect terms

01:09 Identify coefficients

01:24 Use the roots formula

01:48 Substitute appropriate values and solve

02:07 Calculate the square and products

02:14 The square root of 0 is always equal to 0

02:21 When the root equals 0, the equation will have only one solution

02:48 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

Solve the following equation:

$(x+3)^2=2x+5$

Step-by-step solution

To solve the equation $(x+3)^2 = 2x + 5$ , we proceed as follows:

Step 1: Expand the left side. Using the identity $(a+b)^2 = a^2 + 2ab + b^2$ , we find:
$(x+3)^2 = x^2 + 6x + 9$ .
Step 2: Set the equation to zero by moving all terms to one side:
$x^2 + 6x + 9 = 2x + 5$
Subtract $2x + 5$ from both sides:
$x^2 + 6x + 9 - 2x - 5 = 0$
This simplifies to:
$x^2 + 4x + 4 = 0$ .
Step 3: Solve the quadratic equation $x^2 + 4x + 4 = 0$ . Notice this can be factored as:
$(x+2)^2 = 0$ .
Step 4: Solve for $x$ by setting the factor equal to zero:
$x+2 = 0$ .
Thus, $x = -2$ .

Therefore, the solution to the equation $(x+3)^2 = 2x + 5$ is $x = -2$ .

Final Answer

$x=-2$

Key Points to Remember

Essential concepts to master this topic

Expansion Rule: Use $(a+b)^2 = a^2 + 2ab + b^2$ to expand squares
Standard Form: Move all terms to one side: $x^2 + 4x + 4 = 0$
Verification: Substitute $x = -2$ back: $(-2+3)^2 = 2(-2)+5 = 1$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to expand the squared binomial completely
Don't just square the first and last terms like $(x+3)^2 = x^2 + 9$ = missing the middle term! This gives $x^2 - 2x - 4 = 0$ instead of the correct $x^2 + 4x + 4 = 0$ . Always remember the middle term $2ab$ when expanding $(a+b)^2$ .

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

Why do I need to expand $(x+3)^2$ instead of taking the square root?

Taking the square root of both sides only works when the right side is a perfect square or constant. Since we have $2x + 5$ on the right, we must expand and solve as a quadratic equation.

How do I remember the expansion formula $(a+b)^2$ ?

Think "First, Outer, Inner, Last" or use the pattern: square the first term, add twice the product of both terms, then square the last term. For $(x+3)^2$ : $x^2 + 2(x)(3) + 3^2 = x^2 + 6x + 9$ .

What does it mean that $x^2 + 4x + 4 = (x+2)^2$ ?

This is called a perfect square trinomial because it factors into a perfect square! When $(x+2)^2 = 0$ , the only solution is $x = -2$ (called a repeated root).

Could I solve this using the quadratic formula instead?

Yes! Using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ with $a=1, b=4, c=4$ gives $x = \frac{-4 \pm 0}{2} = -2$ . But recognizing the perfect square trinomial is faster!

How can I check if my factoring is correct?

Expand your factored form! If $(x+2)^2$ is correct, then $(x+2)^2 = x^2 + 4x + 4$ , which matches our simplified equation.

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Solve the Quadratic: Expanding (x+3)² to Find x in 2x+5

Quadratic Equations with Perfect Square Trinomials

❤️ 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 do I need to expand $(x+3)^2$ instead of taking the square root?

How do I remember the expansion formula $(a+b)^2$ ?

What does it mean that $x^2 + 4x + 4 = (x+2)^2$ ?

Could I solve this using the quadratic formula instead?

How can I check if my factoring is correct?

🌟 Unlock Your Math Potential

More Questions

The Quadratic Formula

Solve for X in the Quadratic Equation: (x-5)²-5=10+2x

Solve the Quadratic Equation: Finding x in -x² + x - 2 = -2x^2 - 2x - 4

Solve for X in the Fractional Power Equation: ((1/x)+(1/2))² / ((1/x)+(1/3))² = 81/64

Solving the Quadratic: Tackling (x-5)² - 5 = -12 + 2x

Square of sum

Determine Variables A, B, and C for the Equation: (A/X) + (BX/2) = ((2X+3)²)/X - C

Transform the Fraction Equation: Simplifying 1 + y/x + x/4y = 0

Solve the Symmetric Squares Equation: (1 - x)² = (2x + 1)²

Solve the Equation: Finding X in (x+3)² + 2x² = 18

Combining Short Multiplication Formulas

Solve the Quadratic Equation: Find X in 2x^2 - 2x = (x + 1)^2

Solve the Quadratic Equation: Finding X in 7x + 1 + (2x + 3)² = (4x + 2)²

Solve (x-4)² + 3x² = -16x + 12: Multiple Squared Terms Equation

Solve the Quadratic Equation: (x+2)² = (2x+3)²

Solve the Quadratic: Expanding (x+3)² to Find x in 2x+5

Quadratic Equations with Perfect Square Trinomials

❤️ 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 do I need to expand (x+3)2 (x+3)^2 (x+3)2 instead of taking the square root?

How do I remember the expansion formula (a+b)2 (a+b)^2 (a+b)2?

What does it mean that x2+4x+4=(x+2)2 x^2 + 4x + 4 = (x+2)^2 x2+4x+4=(x+2)2?

Could I solve this using the quadratic formula instead?

How can I check if my factoring is correct?

🌟 Unlock Your Math Potential

More Questions

The Quadratic Formula

Solve for X in the Quadratic Equation: (x-5)²-5=10+2x

Solve the Quadratic Equation: Finding x in -x² + x - 2 = -2x^2 - 2x - 4

Solve for X in the Fractional Power Equation: ((1/x)+(1/2))² / ((1/x)+(1/3))² = 81/64

Solving the Quadratic: Tackling (x-5)² - 5 = -12 + 2x

Square of sum

Determine Variables A, B, and C for the Equation: (A/X) + (BX/2) = ((2X+3)²)/X - C

Transform the Fraction Equation: Simplifying 1 + y/x + x/4y = 0

Solve the Symmetric Squares Equation: (1 - x)² = (2x + 1)²

Solve the Equation: Finding X in (x+3)² + 2x² = 18

Combining Short Multiplication Formulas

Solve the Quadratic Equation: Find X in 2x^2 - 2x = (x + 1)^2

Solve the Quadratic Equation: Finding X in 7x + 1 + (2x + 3)² = (4x + 2)²

Solve (x-4)² + 3x² = -16x + 12: Multiple Squared Terms Equation

Solve the Quadratic Equation: (x+2)² = (2x+3)²

Why do I need to expand $(x+3)^2$ instead of taking the square root?

How do I remember the expansion formula $(a+b)^2$ ?

What does it mean that $x^2 + 4x + 4 = (x+2)^2$ ?