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Math  /  Algebra

QuestionSubtract. t2+7t−18t2−6t+5−−6t2+8t−12t2−6t+5t2+7t−18t2−6t+5−−6t2+8t−12t2−6t+5=□\frac{\frac{t^{2}+7 t-18}{t^{2}-6 t+5}-\frac{-6 t^{2}+8 t-12}{t^{2}-6 t+5}}{\frac{t^{2}+7 t-18}{t^{2}-6 t+5}-\frac{-6 t^{2}+8 t-12}{t^{2}-6 t+5}=\square} (Simplify your answer. Use integers or fractions for any numbers in the expression.)

Studdy Solution

STEP 1

1. The problem involves subtracting two rational expressions with a common denominator.
2. Simplification of the expression is required.
3. The expression can be simplified by subtracting the numerators directly since the denominators are the same.

STEP 2

1. Identify and simplify the numerators of the rational expressions.
2. Subtract the simplified numerators.
3. Simplify the resulting expression.

STEP 3

Identify the numerators of the rational expressions:
Numerator 1: t2+7t−18 t^2 + 7t - 18
Numerator 2: −6t2+8t−12 -6t^2 + 8t - 12
The common denominator is t2−6t+5 t^2 - 6t + 5 .

STEP 4

Subtract the numerators:
(t2+7t−18)−(−6t2+8t−12) (t^2 + 7t - 18) - (-6t^2 + 8t - 12)
Distribute the negative sign across the second numerator:
t2+7t−18+6t2−8t+12 t^2 + 7t - 18 + 6t^2 - 8t + 12
Combine like terms:
(t2+6t2)+(7t−8t)+(−18+12) (t^2 + 6t^2) + (7t - 8t) + (-18 + 12)
7t2−t−6 7t^2 - t - 6

STEP 5

The expression simplifies to:
7t2−t−6t2−6t+5 \frac{7t^2 - t - 6}{t^2 - 6t + 5}
Since the numerator and the denominator have no common factors, this is the simplified form of the expression.
The simplified expression is:
7t2−t−6t2−6t+5 \boxed{\frac{7t^2 - t - 6}{t^2 - 6t + 5}}

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