How do I factor x² – 5x – 14 = 0?

To factor the quadratic equation x² – 5x – 14 = 0, we can use the quadratic formula or factoring by grouping. Let’s go with factoring by grouping, which is a simpler method in this case.

Step 1: Write down the equation. x² – 5x – 14 = 0

Step 2: Find two numbers that multiply to the constant term (-14) and add up to the coefficient of the middle term (-5). These two numbers are -7 and +2 because (-7) * (+2) = -14 and (-7) + (+2) = -5.

Step 3: Rewrite the middle term (-5x) using these two numbers: x² – 7x + 2x – 14 = 0

Step 4: Now, group the terms and factor by grouping: (x² – 7x) + (2x – 14) = 0

Step 5: Factor out the greatest common factor (GCF) from each group: x(x – 7) + 2(x – 7) = 0

Step 6: Notice that both groups have a common factor of (x – 7). Factor it out: (x – 7)(x + 2) = 0

Step 7: Now, set each factor equal to zero and solve for x: x – 7 = 0 or x + 2 = 0

Step 8: Solve for x in each equation: For the first equation, x – 7 = 0: x = 7

For the second equation, x + 2 = 0: x = -2

So, the solutions to the equation x² – 5x – 14 = 0 are: x = 7 and x = -2.

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