Question
factor 4𝑥^2+5𝑥−6
Answers
Answered by
GPT 3.5
To factor the quadratic expression 4𝑥^2 + 5𝑥 - 6, we need to find two binomials whose product gives the original expression.
Split the middle term by finding two numbers whose product is equal to the product of the quadratic term coefficient and constant term, which, in this case, is 4 * -6 = -24 and whose sum is equal to the linear term coefficient, which is 5.
The numbers -3 and 8 satisfy these conditions because -3 * 8 = -24 and -3 + 8 = 5.
Now, rewrite the original expression by splitting the middle term using -3𝑥 and 8𝑥:
4𝑥^2 - 3𝑥 + 8𝑥 - 6
Next, factor by grouping:
(4𝑥^2 - 3𝑥) + (8𝑥 - 6)
𝑥(4𝑥 - 3) + 2(4𝑥 - 3)
Now, notice that both terms have a common factor, (4𝑥 - 3):
(𝑥 + 2)(4𝑥 - 3)
Therefore, the fully factored form of 4𝑥^2 + 5𝑥 - 6 is (𝑥 + 2)(4𝑥 - 3).
Split the middle term by finding two numbers whose product is equal to the product of the quadratic term coefficient and constant term, which, in this case, is 4 * -6 = -24 and whose sum is equal to the linear term coefficient, which is 5.
The numbers -3 and 8 satisfy these conditions because -3 * 8 = -24 and -3 + 8 = 5.
Now, rewrite the original expression by splitting the middle term using -3𝑥 and 8𝑥:
4𝑥^2 - 3𝑥 + 8𝑥 - 6
Next, factor by grouping:
(4𝑥^2 - 3𝑥) + (8𝑥 - 6)
𝑥(4𝑥 - 3) + 2(4𝑥 - 3)
Now, notice that both terms have a common factor, (4𝑥 - 3):
(𝑥 + 2)(4𝑥 - 3)
Therefore, the fully factored form of 4𝑥^2 + 5𝑥 - 6 is (𝑥 + 2)(4𝑥 - 3).