Problem 159
Question
Factor. \(4 q^{2}-7 q-2\)
Step-by-Step Solution
Verified Answer
(q - 2)(4q + 1)
1Step 1 - Identify the coefficients
Look at the quadratic expression: \(4q^{2} - 7q - 2\)Identify the coefficients: a = 4, b = -7, c = -2
2Step 2 - Compute ac
Multiply the coefficients a and c:\(4 \times (-2) = -8\)
3Step 3 - Find two numbers that multiply and sum correctly
We need to find two numbers that multiply to -8 (ac) and add to -7 (b). These numbers are -8 and 1 because: \((-8) \times 1 = -8\) and \((-8) + 1 = -7\)
4Step 4 - Rewrite the middle term
Rewrite the quadratic expression by breaking the middle term \(-7q\) into \(-8q + q\): \(4q^{2} - 8q + q - 2\)
5Step 5 - Factor by grouping
Group the terms to factor by grouping:\((4q^{2} - 8q) + (q - 2)\)Factor out the common factors from each group:\(4q(q - 2) + 1(q - 2)\)
6Step 6 - Factor out the common binomial
There is a common binomial \((q - 2)\). Factor this out:\( (q - 2)(4q + 1)\)
Key Concepts
CoefficientsAC MethodFactoring by GroupingQuadratic Expression
Coefficients
To start factoring a quadratic expression, we first need to identify its coefficients. A quadratic expression is typically in the form of \[ ax^2 + bx + c \] In this expression,
- a: the coefficient of \(x^2\)
- b: the coefficient of \(x\)
- c: the constant term
AC Method
The AC Method helps us factor quadratic expressions. We first multiply the coefficients of the \(x^2\) term and the constant term.
In our example:\[ a \times c = 4 \times (-2) = -8 \]
This result, -8, will help us find two numbers that multiply to -8 and add up to b, the coefficient of the linear term (-7). Here:
In our example:\[ a \times c = 4 \times (-2) = -8 \]
This result, -8, will help us find two numbers that multiply to -8 and add up to b, the coefficient of the linear term (-7). Here:
- Numbers are -8 and 1 because:
- \((-8) \times 1 = -8\)
- \((-8) + 1 = -7\)
Factoring by Grouping
Now, using our newfound numbers, we break the middle term \(-7q\) into two terms, \(-8q\) and \(1q\). This gives us:
\[ 4q^{2} - 8q + q - 2 \]
We proceed by grouping terms to factor them easily:
\[ (4q^{2} - 8q) + (q - 2) \]
Next, we factor out the common factor from each group:
\[ 4q(q - 2) + 1(q - 2) \]
We factor out the common binomial factor \((q - 2)\) from both groups.
\[ 4q^{2} - 8q + q - 2 \]
We proceed by grouping terms to factor them easily:
\[ (4q^{2} - 8q) + (q - 2) \]
Next, we factor out the common factor from each group:
- Common factor of the first group: \(4q\), so \(4q(q - 2)\)
- Common factor of the second group: \(1\), so \(1(q - 2)\)
\[ 4q(q - 2) + 1(q - 2) \]
We factor out the common binomial factor \((q - 2)\) from both groups.
Quadratic Expression
A quadratic expression is any expression that can be written in the form: \[ ax^2 + bx + c \]
This is a polynomial of degree 2 because the highest power of the variable (in this case, \(x\)) is 2. In our example \(4q^{2} - 7q - 2\), '4q²' indicates that it's a quadratic expression.
Factoring quadratic expressions helps solve quadratic equations. The last step of our example:
\[ 4q(q - 2) + 1(q - 2)\rightarrow(q - 2)(4q + 1)\]
shows how we simplified the quadratic expression into a factored form. Understanding how to work with quadratic expressions is essential for solving quadratic equations efficiently.
This is a polynomial of degree 2 because the highest power of the variable (in this case, \(x\)) is 2. In our example \(4q^{2} - 7q - 2\), '4q²' indicates that it's a quadratic expression.
Factoring quadratic expressions helps solve quadratic equations. The last step of our example:
\[ 4q(q - 2) + 1(q - 2)\rightarrow(q - 2)(4q + 1)\]
shows how we simplified the quadratic expression into a factored form. Understanding how to work with quadratic expressions is essential for solving quadratic equations efficiently.