Question

quadratic equations
a quadratic equation is an equation of the form:
y = ax² + bx + c standard form
where a, b, and c are constants
factoring quadratic expression
examples:
① x² + 3x
x·x + 3·x
x(x + 3)
② 4x² - 32x
4·x·x - 4·8·x
4x(x - 8)
③ x² - 4x - 5
(x - 5)(x + 1)
④ x² - 13x + 36

Explanation:

To factor the quadratic expression \( x^2 - 13x + 36 \), we follow these steps:

Step 1: Identify \( a \), \( b \), and \( c \)

For the quadratic \( x^2 - 13x + 36 \), we have:

• \( a = 1 \) (coefficient of \( x^2 \))
• \( b = -13 \) (coefficient of \( x \))
• \( c = 36 \) (constant term)

Step 2: Find two numbers that multiply to \( a \cdot c \) and add to \( b \)

We need two numbers that:

• Multiply to \( 1 \cdot 36 = 36 \)
• Add to \( -13 \)

Let's list the factor pairs of \( 36 \):

• \( 1 \) and \( 36 \) (sum: \( 37 \))
• \( 2 \) and \( 18 \) (sum: \( 20 \))
• \( 3 \) and \( 12 \) (sum: \( 15 \))
• \( 4 \) and \( 9 \) (sum: \( 13 \))
• \( -4 \) and \( -9 \) (sum: \( -13 \))

Ah, \( -4 \) and \( -9 \) work because:

• \( (-4) \cdot (-9) = 36 \)
• \( (-4) + (-9) = -13 \)

Step 3: Rewrite the middle term using these two numbers

We rewrite \( -13x \) as \( -4x - 9x \):

$$ x^2 - 4x - 9x + 36 $$

Step 4: Factor by grouping

Group the first two terms and the last two terms:

$$ (x^2 - 4x) + (-9x + 36) $$

Factor out the greatest common factor (GCF) from each group:

• From \( x^2 - 4x \), the GCF is \( x \): \( x(x - 4) \)
• From \( -9x + 36 \), the GCF is \( -9 \): \( -9(x - 4) \)

So now we have:

$$ x(x - 4) - 9(x - 4) $$

Step 5: Factor out the common binomial factor

Notice that both terms have a common factor of \( (x - 4) \). Factor that out:

$$ (x - 4)(x - 9) $$

Final Answer

The factored form of \( x^2 - 13x + 36 \) is \(\boxed{(x - 4)(x - 9)}\).

Answer:

To factor the quadratic expression \( x^2 - 13x + 36 \), we follow these steps:

Step 1: Identify \( a \), \( b \), and \( c \)

For the quadratic \( x^2 - 13x + 36 \), we have:

• \( a = 1 \) (coefficient of \( x^2 \))
• \( b = -13 \) (coefficient of \( x \))
• \( c = 36 \) (constant term)

Step 2: Find two numbers that multiply to \( a \cdot c \) and add to \( b \)

We need two numbers that:

• Multiply to \( 1 \cdot 36 = 36 \)
• Add to \( -13 \)

Let's list the factor pairs of \( 36 \):

• \( 1 \) and \( 36 \) (sum: \( 37 \))
• \( 2 \) and \( 18 \) (sum: \( 20 \))
• \( 3 \) and \( 12 \) (sum: \( 15 \))
• \( 4 \) and \( 9 \) (sum: \( 13 \))
• \( -4 \) and \( -9 \) (sum: \( -13 \))

Ah, \( -4 \) and \( -9 \) work because:

• \( (-4) \cdot (-9) = 36 \)
• \( (-4) + (-9) = -13 \)

Step 3: Rewrite the middle term using these two numbers

We rewrite \( -13x \) as \( -4x - 9x \):

$$ x^2 - 4x - 9x + 36 $$

Step 4: Factor by grouping

Group the first two terms and the last two terms:

$$ (x^2 - 4x) + (-9x + 36) $$

Factor out the greatest common factor (GCF) from each group:

• From \( x^2 - 4x \), the GCF is \( x \): \( x(x - 4) \)
• From \( -9x + 36 \), the GCF is \( -9 \): \( -9(x - 4) \)

So now we have:

$$ x(x - 4) - 9(x - 4) $$

Step 5: Factor out the common binomial factor

Notice that both terms have a common factor of \( (x - 4) \). Factor that out:

$$ (x - 4)(x - 9) $$

Final Answer

The factored form of \( x^2 - 13x + 36 \) is \(\boxed{(x - 4)(x - 9)}\).