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Course: Digital SAT Math>Unit 8

Lesson 7: Solving quadratic equations: medium

Solving quadratic equations — Basic example

Watch Sal work through a basic Solving quadratic equations problem.

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• Wait, when Sal does the 4x^2= 52 example, he says that the solution is +√13 and -√13. If we plug in the -√13 into the original quadratic equation, doesn't the negative sign remain because it is outside the square root sign? So, isn't -√13 not a solution to the quadratic equation?
• When you plug in a value for x, it behaves as if it were contained within parentheses for the purpose of order of operations. So 4x^2=52 becomes 4(-√13)^2 = 52, which in turn means 4 (-√13)(-√13) = 52. Since you square the entire -√13 expression, the negative cancels itself out, leaving you with 13.

What you are thinking about is something like -3^2, which is in fact -9, because order of operations suggests it is actually (-1)(3)^2 = (-1)(3)(3) = (-1)(9) = -9. If you were plugging -3 into an equation to replace x, such as x^2, then it would be evaluated as (-3)^2, or simply, 9.
• Why positive or negative in the square root?
x² = 9
we know that something times itself is 9
We know immediately that x can equal 3, because 3 x 3 = 9
However, we have to put on the brakes here, because there is another way to get 9 by multiplying negative 3 by itself:
-3 x -3 = 9
Because of that, if we are solving x² = 9, we have to allow for either correct answer.
So we say, x = ± 3 and that means that x = 3 or x = -3

When we have the more complicated case of x² = 13
the square root will be x = ± √13 and that means we have two possible answers:
x = +√13 and x = - √13

You have to be careful, though, because if the problem says
What is √16 , this only has one answer. √16 means give me the principal square root of 16 and that is asking for only the positive root.
• Please Sal, am finding it hard to solve the word problem questions, even after watching videos examples which i find explanatory, i just keep getting wrong answers during practice. please help me.
• Why cant i just wake up with a 1500+ score
• Because happiness has to be fought for.
• I thought you can't have a negative under the radical. I don't see how you can say "the square root of negative 13?"
• It's not "the square root of negative 13", it's "the negative square root of 13".

For example, if you have 4, its positive square root is 2, its negative square root is -2.
• When you say the sqrt of 13 aren't you allowing for both the negative and positive already?
• Wait, when Sal does the 4x^2= 52 example, he says that the solution is +√13 and -√13. If we plug in the -√13 into the original quadratic equation, doesn't the negative sign remain because it is outside the square root sign? So, isn't -√13 not a solution to the quadratic equation?
• It is a solution because ( -√13 )^2 is 13 and 13*4 =52.
• does equations that have x^2 always have 2 answers, what about ones with just x
• You right! The equation x^2 would have two answers, because when you have something that is square rooted it has a positive and a negative root. X would only have one answer because it would be what ever X was. Try going to a online graphing calculator called Desmos and plug those in and you will see that it crosses the x axis twice for the first one and once or the second one.