The Quadratic Formula: Factoring trinomials using decomposition

The quadratic equation given is:

$x^2 + 7\frac{3}{4}x - 2 = 0$

First, let's rewrite the coefficient $b$ as a decimal for simplicity:

$b = 7\frac{3}{4} = 7.75$

Given $a = 1$, $b = 7.75$, and $c = -2$, we apply the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Calculate the discriminant:

$\text{Discriminant} = b^2 - 4ac = (7.75)^2 - 4 \times 1 \times -2$

$= 60.0625 + 8 = 68.0625$

Now, find the roots by plugging into the quadratic formula:

$x = \frac{-7.75 \pm \sqrt{68.0625}}{2}$

Calculate the square root:

$\sqrt{68.0625} = 8.25$

Find the roots:

$x_1 = \frac{-7.75 + 8.25}{2} = \frac{0.5}{2} = 0.25$

$x_2 = \frac{-7.75 - 8.25}{2} = \frac{-16}{2} = -8$

Thus, the factored form of the trinomial can be written using these roots:

$(x - x_1)(x - x_2) = (x - 0.25)(x + 8)$

Rewriting with fractional representations:

$(x - \frac{1}{4})(x + 8)$

Thus, the trinomial can be expressed as:

$(x + 8)(x - \frac{1}{4})$

Comparing with the options provided, the correct answer is:

$\boxed{(x + 8)(x - \frac{1}{4})}$

To solve the problem of factoring the quadratic expression $5x^2 + 9x + 4$, we can use the decomposition method:

• Step 1: Identify $a = 5$, $b = 9$, and $c = 4$. Calculate the product $a \cdot c = 5 \cdot 4 = 20$.
• Step 2: We need two numbers whose product is 20 and whose sum is 9. The numbers 5 and 4 satisfy this condition because $5\times4=20$ and $5+4=9$.
• Step 3: Decompose the middle term using these numbers: $5x^2 + 5x + 4x + 4$.
• Step 4: Group the terms: $(5x^2 + 5x) + (4x + 4)$.
• Step 5: Factor each group: $5x(x + 1) + 4(x + 1)$.
• Step 6: Use the distributive property to factor out the common binomial: $(5x + 4)(x + 1)$.

Therefore, the factorization of the expression $5x^2 + 9x + 4$ is $(5x+4)(x+1)$.

To solve this problem, we must factor the expression $x^2 - 2bx + ax - 2ba + 7x - 14b$. Let’s go through the steps involved:

• Step 1: Rearrange the terms to enable easier grouping: $x^2 + (ax + 7x) + (-2bx - 2ba - 14b)$.
• Step 2: Group the terms in pairs that can be factored individually:
  • First Group: $x^2 + (a+7)x$
  • Second Group: $-2b(x + a) - 14b$
• Step 3: Look for common factors for each group. Factor them out:
  • First Group: Factor out $x$, giving $x(x + a + 7)$
  • Second Group: Factor out $-2b$, giving $-2b(x + a + 7)$
• Step 4: Factor by grouping common terms.
  Both the expressions now have a common factor $(x + a + 7)$, so we can express the original expression as: $(x - 2b)(x + a + 7)$.
• Step 5: Simplify and ensure correctness: Correctly reordered the terms within the parentheses as necessary to reflect the correct factorization.

Therefore, the expression $x^2 - 2bx + ax - 2ba + 7x - 14b$ can be factored as $(x+7+a)(x-2b)$.