Question Video: Forming a Quadratic Equation in the Simplest Form Using the Relation between the Quadratic Equation and Its Roots | Nagwa Question Video: Forming a Quadratic Equation in the Simplest Form Using the Relation between the Quadratic Equation and Its Roots | Nagwa

# Question Video: Forming a Quadratic Equation in the Simplest Form Using the Relation between the Quadratic Equation and Its Roots Mathematics • First Year of Secondary School

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Given that ๐ฟ and ๐ are the roots of the equation 3๐ฅยฒ + 16๐ฅ โ 1 = 0, find, in its simplest form, the quadratic equation whose roots are ๐ฟ/2 and ๐/2.

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### Video Transcript

Given that ๐ฟ and ๐ are the roots of the equation three ๐ฅ squared plus 16๐ฅ minus one equals zero, find, in its simplest form, the quadratic equation whose roots are ๐ฟ over two and ๐ over two.

Letโs begin by recalling the relationship between the quadratic equation and its roots. For a quadratic equation whose leading coefficient is one, we can say that the coefficient of ๐ฅ is the negative sum of the roots of the equation, whereas the constant is the product of its roots. Now, comparing this to our equation, we see we do have a bit of a problem. The leading coefficient, the coefficient of ๐ฅ squared, is three. And so, weโre going to divide every single term by three. Three ๐ฅ squared divided by three is ๐ฅ squared. 16๐ฅ divided by three is 16๐ฅ over three or 16 over three ๐ฅ. And then, our constant term becomes negative one-third.

And so, comparing this to the general form, we know that the sum of the roots of our equation is the negative coefficient of ๐ฅ. So, itโs negative 16 over three. Then, the product is negative one-third. But weโre also told that ๐ฟ and ๐ are the roots of our equation, so we can replace these with ๐ฟ plus ๐ as the sum and ๐ฟ times ๐ as the product. Weโre looking to find the quadratic equation whose roots are ๐ฟ over two and ๐ over two. And so, what weโre going to do is find an expression for the sum of these roots; thatโs ๐ฟ over two plus ๐ over two. Similarly, weโre going to find the product of these roots; thatโs ๐ฟ over two times ๐ over two which is ๐ฟ๐ over four.

So, how can we link the equations we have? Well, letโs call this first equation one. We have ๐ฟ plus ๐ as being equal to negative 16 over three. If we divide the entire expression by two, that is ๐ฟ plus ๐ over two, we actually know thatโs equal to ๐ฟ over two plus ๐ over two. And so, this means we can find the value of ๐ฟ over two plus ๐ over two by dividing the value for ๐ฟ plus ๐ by two. Thatโs negative 16 over three divided by two, which is negative eight over three.

And we can repeat this process with our second equation. This time, of course, we want ๐ฟ๐ divided by four. So thatโs going to be negative one-third divided by four, which is negative one twelfth. And now that we have the sum of our roots and the product, we can substitute these back into our earlier equation. When we do, we find that the quadratic equation whose roots are ๐ฟ over two and ๐ over two is ๐ฅ squared minus negative eight over three ๐ฅ plus negative one twelfth equals zero.

Letโs simplify a little by dealing with our signs. In other words, negative negative eight-thirds is just eight-thirds. And adding negative one twelfth is the same as subtracting one twelfth. Our final step is to create integer coefficients. And to do so, weโre going to multiply through by 12. ๐ฅ squared times 12 is 12๐ฅ squared. Then, if we multiply eight-thirds by 12, we cancel a three. And we end up working out eight times four, which is 32. Negative one twelfth times 12 is negative one. And, of course, zero times 12 is zero. And so, the quadratic equation is 12๐ฅ squared plus 32๐ฅ minus one equals zero.

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