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

Given that 𝐿 and 𝑀 are the roots of the equation 2π‘₯Β² βˆ’ 10π‘₯ + 1 = 0, find, in its simplest form, the quadratic equation whose roots are 𝐿/3 and 𝑀/3.

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Video Transcript

Given that 𝐿 and 𝑀 are the roots of the equation two π‘₯ squared minus 10π‘₯ plus one equals zero, find, in its simplest form, the quadratic equation whose roots are 𝐿 over three and 𝑀 over three.

We begin by recalling some key facts about a quadratic equation written 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 two roots of the quadratic are π‘Ÿ sub one and π‘Ÿ sub two, their sum is equal to negative 𝑏 over π‘Ž. The product of the two roots, π‘Ÿ one multiplied by π‘Ÿ two, is equal to 𝑐 over π‘Ž.

In this question, we are given the equation two π‘₯ squared minus 10π‘₯ plus one equals zero. The values of π‘Ž, 𝑏, and 𝑐 here are two, negative 10, and one. We are also told that 𝐿 and 𝑀 are the two roots of the equation. Therefore, the sum of these, 𝐿 plus 𝑀, is equal to negative negative 10 over two. This is equal to five. The product of the roots 𝐿 multiplied by 𝑀 is equal to one over two or one-half. We can now use these values to help find the quadratic equation whose roots are 𝐿 over three and 𝑀 over three.

Let’s begin by considering the sum of these roots. We have 𝐿 over three plus 𝑀 over three. As the denominators are the same, we can simply add the numerators, giving us 𝐿 plus 𝑀 over three. We have already worked out that 𝐿 plus 𝑀 is equal to five. This means that the sum of the roots is equal to five over three or five-thirds. Negative 𝑏 over π‘Ž is equal to five-thirds.

We will now consider the product of our two roots. We need to multiply 𝐿 over three and 𝑀 over three. We do this by multiplying the numerators and denominators separately, giving us 𝐿𝑀 over nine. As 𝐿𝑀 is equal to one-half, we have a half over nine or a half divided by nine. This is equal to one eighteenth. The product of our roots 𝑐 over π‘Ž is equal to one eighteenth.

We can now solve these two equations to find the values of π‘Ž, 𝑏, and 𝑐. We note that the denominators on the right-hand side are different. However, on the left-hand side of both our equations, the denominator is π‘Ž. We can therefore multiply the numerator and denominator of the right-hand side of our first equation by six. Five over three or five-thirds is the same as 30 over 18.

Our next step is to let π‘Ž equal 18. From the first equation, this means that negative 𝑏 is equal to 30, which means that 𝑏 is equal to negative 30. From the second equation, if 𝑐 over π‘Ž is equal to one over 18 and π‘Ž equals 18, then 𝑐 must be equal to one. The quadratic equation whose roots are 𝐿 over three and 𝑀 over three is 18π‘₯ squared minus 30π‘₯ plus one equals zero.

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