Question Video: Determining the Type of Roots of a Quadratic Equation Mathematics

How many real roots does the equation 6π‘₯Β² + 7π‘₯ βˆ’ 7 = 0 have?

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Video Transcript

How many real roots does the equation six π‘₯ squared plus seven π‘₯ minus seven equals zero have?

Well, the first thing to notice is that we have an equation here in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 equals zero. And our equation is a quadratic equation. Well, if we want to evaluate the roots, what we can use is something called the discriminant. And the discriminant is something which can be found using 𝑏 squared minus four π‘Žπ‘, where 𝑏 is the coefficient of π‘₯, π‘Ž is the coefficient of π‘₯ squared, and 𝑐 is a numerical value.

And why is this useful? Well, it’s useful because it can help us to analyze our roots. So first of all, if 𝑏 squared minus four π‘Žπ‘ is greater than zero, then we know that our roots are gonna be real and different, which means that we’re gonna have two roots, whereas if 𝑏 squared minus four π‘Žπ‘, so our discriminant, is equal to zero, our roots is gonna be real and the same, which means that we’re gonna have one repeated root. And if the discriminant is less than zero, our roots are gonna be not real and complex. So therefore, we can say they’re gonna be no real roots.

Okay, great. So we now know what the discriminant is gonna be used for. So let’s use this to find out how many real roots our equation has. Well, for six π‘₯ squared plus seven π‘₯ minus seven equals zero, our π‘Ž is gonna be equal to six, our 𝑏 is equal to seven, and our 𝑐 is equal to negative seven cause remembering that the signs are important. So we have to consider these. So we’ve got 𝑐 is equal to negative seven.

So therefore, if we want to find the discriminant, it’s gonna be equal to seven squared minus four multiplied by six multiplied by negative seven, which is gonna be equal to 49 minus negative 168, which is gonna be equal to 217 because if we subtract a negative, it’s the same as adding a positive. So therefore, we can see that our discriminant is greater than zero, so we know that the roots are gonna be real and different. So in answer to the question, how many real roots does the equation six π‘₯ squared plus seven π‘₯ minus seven equals zero have, the answer is two roots.

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