Question Video: Forming a Quadratic Equation in the Simplest Form given Its Roots Mathematics

If 𝐿 and 𝑀 are the roots of the equation π‘₯Β² + 17π‘₯ + 1 = 0, find, in its simplest form, the quadratic equation whose roots are 3𝐿 and 3𝑀.

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Video Transcript

If 𝐿 and 𝑀 are the roots of the equation π‘₯ squared plus 17π‘₯ plus one equals zero, find, in its simplest form, the quadratic equation whose roots are three 𝐿 and three 𝑀.

Let’s begin by recalling some information about the quadratic equation in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 equals zero, where π‘Ž, 𝑏, and 𝑐 are constants and π‘Ž is nonzero. If the roots of this equation are π‘Ÿ sub one and π‘Ÿ sub two, the sum of these roots is equal to negative 𝑏 over π‘Ž. The product of the roots π‘Ÿ one multiplied by π‘Ÿ two is equal to 𝑐 over π‘Ž.

We are given the equation π‘₯ squared plus 17π‘₯ plus one is equal to zero. This means that the values of π‘Ž, 𝑏, and 𝑐 are one, 17, and one, respectively. We are also told that the roots of this equation are 𝐿 and 𝑀. The sum of these roots 𝐿 plus 𝑀 is therefore equal to negative 17 over one. This is equal to negative 17. The product of the roots 𝐿 multiplied by 𝑀 is equal to one over one. This is equal to one. We now need to find the quadratic equation whose roots are three 𝐿 and three 𝑀.

Let’s begin by considering the sum of these roots. This is equal to three 𝐿 plus three 𝑀. And these terms have a common factor of three. We can, therefore, rewrite this as three multiplied by 𝐿 plus 𝑀. As 𝐿 plus 𝑀 is equal to negative 17, we need to multiply three by negative 17. This is equal to negative 51. In our new quadratic equation, negative 𝑏 over π‘Ž is equal to negative 51.

Let’s now consider the product of our roots. We need to multiply three 𝐿 by three 𝑀. This simplifies to nine 𝐿𝑀. And as 𝐿𝑀 is equal to one, we have nine multiplied by one. 𝑐 over π‘Ž is, therefore, equal to nine.

We now have two equations that we can solve to calculate the values of π‘Ž, 𝑏, and 𝑐. As negative 51 and nine are integers, we can let π‘Ž equal one. From the first equation, this means that negative 𝑏 is equal to negative 51, which in turn means that 𝑏 is equal to 51. From the second equation, if π‘Ž is equal to one, 𝑐 is equal to nine.

The quadratic equation whose roots are three 𝐿 and three 𝑀 is π‘₯ squared plus 51π‘₯ plus nine equals zero.

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