To factor the quadratic expression \( x^2 + 4x - 21 \), we can follow these steps:
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Identify the coefficients:
- The coefficient of \( x^2 \) (which is \( a \)) is 1.
- The coefficient of \( x \) (which is \( b \)) is 4.
- The constant term (which is \( c \)) is -21.
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Factor the expression: We are looking for two numbers that multiply to \( ac = 1 \times -21 = -21 \) and add to \( b = 4 \).
The two numbers that meet these conditions are \( 7 \) and \( -3 \) because:
- \( 7 \times -3 = -21 \)
- \( 7 + (-3) = 4 \)
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Rewrite the middle term: We can rewrite the expression as: \[ x^2 + 7x - 3x - 21 \]
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Group the terms: Now, group the terms: \[ (x^2 + 7x) + (-3x - 21) \]
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Factor by grouping: Factor out the common factors from each group: \[ x(x + 7) - 3(x + 7) \]
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Factor out the common binomial: Now, we can factor out \( (x + 7) \): \[ (x + 7)(x - 3) \]
So, the factored form of \( x^2 + 4x - 21 \) is: \[ (x + 7)(x - 3) \]