lesson
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If someone asks you for a number that multiplies by itself to make 25, you will probably shout out 5. But there is a secret twin hiding in plain sight: (−5)×(−5) also equals 25.
A quadratic equation is any equation where the highest exponent on the variable is 2, such as x2=25. Whenever you undo a square by taking a square root, you must account for both the positive and negative answers.
The Square Root Property
The Square Root Property states that if x2=k and k≥0, then x=k or x=−k, which we write compactly as x=±k.
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Why does this work? If you subtract k to get x2−k=0, this factors as a difference of squares into (x−k)(x+k)=0, giving both solutions directly.
How do we handle equations where x2 is multiplied by a number or shifted by a constant? Let's isolate x2 first.
Solving Equations of the Form ax² = c
To solve ax2=c, your first goal is to isolate the squared term by dividing both sides by the coefficient a, which is the numerical factor multiplying the variable.
Let's solve 3x2−48=0. First, add 48 to both sides to get 3x2=48, then divide by 3 to get x2=16.
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Not every square root simplifies to an integer. If you solve 2x2=14, dividing by 2 gives x2=7, leading to the exact irrational roots x=±7.
A common mistake is writing only the positive root x=7 and forgetting the negative root −7. Always check that you wrote the plus-or-minus symbol ± the exact moment you take the square root of both sides.