Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.
When we solved linear equations, if an equation had too many fractions we ‘cleared the fractions’ by multiplying both sides of the equation by the LCD.
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This property may seem fairly obvious, but it has big implications for solving quadratic equations.
If you have a factored polynomial that is equal to 0, you know that at least one of the factors or both factors equal 0.
We cannot take the square root of a negative number.
So, when we substitute , , and into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. The quadratic equations we have solved so far in this section were all written in standard form, .
The two values that we found via factoring, x = −4 and x = 3, lead to true statements: 0 = 0. But x = 5, the value not found by factoring, creates an untrue statement—27 does not equal 0!
Note in the example above, if the common factor of 2 had been factored out, the resulting factor would be (−r 3), which is the negative of (r – 3).