Question
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Tutor-Verified Answer
Answer
The factored form of \( 5a^{2} - 23a + 12 \) is \( (a - 4)(5a - 3) \).
Solution
It seems like you want to **factorize** the quadratic expression:
\[
5a^{2} - 23a + 12
\]
Let's factorize it step by step.
### Step 1: Identify the Coefficients
The quadratic expression is in the form:
\[
ax^{2} + bx + c
\]
where:
- \( a = 5 \)
- \( b = -23 \)
- \( c = 12 \)
### Step 2: Find Two Numbers That Multiply to \( a \times c \) and Add to \( b \)
We need two numbers that:
- Multiply to \( 5 \times 12 = 60 \)
- Add up to \( -23 \)
After examining the factors of 60, we find:
\[
-3 \times -20 = 60 \quad \text{and} \quad -3 + (-20) = -23
\]
### Step 3: Rewrite the Middle Term Using These Numbers
\[
5a^{2} - 3a - 20a + 12
\]
### Step 4: Factor by Grouping
Group the terms:
\[
(5a^{2} - 3a) + (-20a + 12)
\]
Factor out the greatest common factor (GCF) from each group:
\[
a(5a - 3) - 4(5a - 3)
\]
### Step 5: Factor Out the Common Binomial Factor
\[
(a - 4)(5a - 3)
\]
### Final Factorized Form
\[
5a^{2} - 23a + 12 = (a - 4)(5a - 3)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To factor the quadratic expression \( 5a^2 - 23a + 12 \), we seek two numbers that multiply to \( 5 \times 12 = 60 \) and add to \(-23\). These numbers are \(-20\) and \(-3\). We can rewrite the middle term using these numbers: \( 5a^2 - 20a - 3a + 12 \). Next, group the terms: \( (5a^2 - 20a) + (-3a + 12) \). Factoring each group gives: \( 5a(a - 4) - 3(a - 4) \). Now, we can factor out the common binomial: \( (5a - 3)(a - 4) \). Thus, the factored form of the quadratic \( 5a^2 - 23a + 12 \) is \( (5a - 3)(a - 4) \). Now you can easily solve for the roots by setting each factor to zero!
