Questions: Add. [ frac64 x^2-23 x+28+frac24 x^2-11 x+7 ] Simplify your answer as much as possible.

$Add. [ frac64 x^2-23 x+28+frac24 x^2-11 x+7 ] Simplify your answer as much as possible.$

Transcript text: Add. \[ \frac{6}{4 x^{2}-23 x+28}+\frac{2}{4 x^{2}-11 x+7} \] Simplify your answer as much as possible.

Solution

Solution Steps

To add the given fractions, we need to find a common denominator. The denominators are quadratic expressions, so we will factor them first. Once factored, we can determine the least common denominator (LCD) and rewrite each fraction with this common denominator. Finally, we will add the numerators and simplify the resulting expression if possible.

Step 1: Factor the Denominators

We start with the denominators $ 4x^2 - 23x + 28 $ and $ 4x^2 - 11x + 7 $. Factoring these gives us: \[ 4x^2 - 23x + 28 = (x - 4)(4x - 7) \] \[ 4x^2 - 11x + 7 = (x - 1)(4x - 7) \]

Step 2: Find the Least Common Denominator (LCD)

The least common denominator (LCD) of the two fractions is: \[ \text{LCD} = (x - 4)(x - 1)(4x - 7)^2 \]

Step 3: Rewrite Each Fraction with the Common Denominator

We rewrite the fractions: \[ \frac{6}{4x^2 - 23x + 28} = \frac{6 \cdot (x - 1)(4x - 7)}{(x - 4)(x - 1)(4x - 7)^2} \] \[ \frac{2}{4x^2 - 11x + 7} = \frac{2 \cdot (x - 4)(4x - 7)}{(x - 4)(x - 1)(4x - 7)^2} \]

Step 4: Add the Fractions

Now we can add the two fractions: \[ \frac{6(x - 1)(4x - 7) + 2(x - 4)(4x - 7)}{(x - 4)(x - 1)(4x - 7)^2} \]

Step 5: Simplify the Result

After simplifying the numerator, we find: \[ \text{Numerator} = 2(4x^2 - 14x + 19) \] Thus, the final simplified expression is: \[ \frac{2(4x^2 - 14x + 19)}{(x - 4)(x - 1)(4x - 7)^2} \]