Question Video: Finding a Root of a Quadratic Equation given the Other Root Mathematics • First Year of Secondary School

Video Transcript

Given that negative 10 is a root of the equation two 𝑥 squared plus 13𝑥 minus 70 equals zero, what is the other root?

We’re told that negative 10 is a root of our equation, which means that our quadratic must be equal to zero when 𝑥 is equal to negative 10. Essentially, it’s a solution to the equation two 𝑥 squared plus 13𝑥 minus 70 equals zero. Now, what this actually means is that 𝑥 plus 10 must be a factor of two 𝑥 squared plus 13𝑥 minus 70. Two 𝑥 squared plus 13𝑥 minus 70 can therefore be written as 𝑥 plus 10 times some other binomial.

Now, let’s give that binomial a form. Let’s say it’s in the form 𝑎𝑥 plus 𝑏, where 𝑎 and 𝑏 are real constants. What we’re going to do is distribute the parentheses on the right-hand side of this equation and see what we get. We begin by multiplying 𝑥 by 𝑎𝑥. That’s 𝑎𝑥 squared. We then multiply the outer terms. That’s 𝑥 times 𝑏, which is 𝑏𝑥. Next, we multiply the inner terms. That’s 10 times 𝑎𝑥, which is 10𝑎𝑥. And finally, we multiply 10 by 𝑏, to give us 10𝑏. So we find that this is equal to two 𝑥 squared plus 13𝑥 minus 70.

And we now use a process called comparing coefficients. We look at the coefficients of our various terms. Let’s begin by comparing our coefficients of 𝑥 squared. On the left-hand side, we have a two. And on the right-hand side, the coefficient of 𝑥 squared is 𝑎. So when we compare coefficients of 𝑥 squared, we find 𝑎 is equal to two.

We could next compare coefficients of 𝑥. In fact though, we’re going to compare constants. We might say these are the coefficients of the 𝑥 to the power of zero terms. On the left-hand side, our constant is negative 70. And on the right-hand side, we have 10𝑏. So negative 70 equals 10𝑏. And so we’re going to divide by 10 to solve for 𝑏. 𝑏 is therefore equal to negative seven. This means that our quadratic expression can be written as 𝑥 plus 10 times two 𝑥 minus seven. We’ve replaced 𝑎 and 𝑏 with their solutions.

But we know that we’re using this to solve the equation two 𝑥 squared plus 13𝑥 minus 70 equals zero. We already know that we have one root of negative 10. That’s found by setting 𝑥 plus 10 equal to zero. We’re now going to set two 𝑥 minus seven equal to zero and solve for 𝑥. We’ll add seven to both sides of this equation so that two 𝑥 is equal to seven. And then we’ll divide through by two. So 𝑥 is equal to seven over two or 3.5. And we could check this solution by substituting 𝑥 equals seven over two into our original equation, making sure that it is indeed equal to zero.