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Answer
\( 3x^{2} + 20x - 7 = (3x - 1)(x + 7) \)
Solution
We need to factor the quadratic polynomial: 3x² + 20x - 7.
Step 1: Multiply the coefficient of x² (which is 3) by the constant term (-7): 3 × (-7) = -21.
Step 2: We need two numbers that multiply to -21 and add up to 20. The numbers 21 and -1 work because 21 × (-1) = -21 and 21 + (-1) = 20.
Step 3: Rewrite the middle term 20x using 21 and -1:
3x² + 21x - x - 7.
Step 4: Factor by grouping:
Group the terms: (3x² + 21x) + (-x - 7).
Factor out common factors in each group:
From 3x² + 21x, factor out 3x: 3x(x + 7).
From -x - 7, factor out -1: -1(x + 7).
Step 5: Notice the common factor (x + 7):
(3x - 1)(x + 7).
Thus, the completely factored form of the expression is (3x - 1)(x + 7).
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Simplify this solution Bonus Knowledge
To factor the quadratic expression \( 3x^{2}+20x-7 \) completely, start by looking for two numbers that multiply to \( 3 \times (-7) = -21 \) and add to \( 20 \). Those numbers are \( 21 \) and \( -1 \). Now, rewrite the middle term: \[ 3x^{2} + 21x - 1x - 7 \] Next, group the terms: \[ (3x^{2} + 21x) + (-1x - 7) \] Factor out the common factors in each group: \[ 3x(x + 7) - 1(x + 7) \] Now, factor out the common binomial: \[ (3x - 1)(x + 7) \] So, the completely factored form is: \[ (3x - 1)(x + 7) \]
