Question Video: Solving Quadratic Equations Using the Quadratic Formula Mathematics

Video Transcript

Find the solution set of the equation five 𝑥 squared minus seven 𝑥 minus 32 equals zero, giving values to three decimal places.

So, we have the quadratic equation and we’re asked to find its solution set, which is just another way of saying solve the equation. We’re told that we need to give values to three decimal places. So, this is a big clue that the quadratic equation we’ve been given won’t be able to be solved by factoring. And so, we need to use another method. This is where we need to use the quadratic formula, which we should make sure we learn off by heart. The quadratic formula tells us that if we have the general quadratic equation 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero. Then the solution set or roots of this equation, if they exist, are given by 𝑥 equals negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 all over two 𝑎.

In order to apply this formula then, what we need to do are determine the values of 𝑎, 𝑏, and 𝑐. 𝑎 is the coefficient of 𝑥 squared. So, for our equation, that’s five. 𝑏 is the coefficient of 𝑥. So, for our equation, that’s negative seven. And we must make sure that we include the negative sign. 𝑐 is the constant term. In our equation, that’s negative 32. Now, all we need to do is substitute these values of 𝑎, 𝑏, and 𝑐 into the quadratic formula. So, we have 𝑥 equals negative 𝑏 — that’s negative negative seven — plus or minus the square root of 𝑏 squared, so negative seven squared, minus four 𝑎𝑐 — that’s minus four times five times negative 32. And this is all over two 𝑎. That’s two times five. You can break the formula down into stages if you wish, or you can substitute straight into the formula. But you must be careful.

Let’s now look at simplifying. We have negative negative seven. So that’s seven. And in the denominator, two times five is 10. Let’s now look at the square root. Firstly, we have negative seven squared, and negative seven squared is equal to 49. If you’re using a calculator to help with this, it’s really common to make a mistake when finding this value. Because what a lot of people do by mistake is they type negative seven squared into their calculator, exactly as I’ve written in orange here.

Now, when you do this, your calculator will give negative 49, but that doesn’t mean your calculator has made a mistake. It means you’ve made a mistake in the question you’ve asked it. Negative seven squared means negative one multiplied by seven squared. And if you recall BIDMAS or the order of operations, we work our indices or powers before multiplication. So, negative seven squared means negative one multiplied by seven squared, which is negative one multiplied by 49, giving negative 49. What we should’ve typed into our calculator was negative seven enclosed in brackets squared. So that ensures that our calculator is taking a negative value first and then squaring it. This is a really common mistake, and one you need to make sure you watch out for.

Continuing under the square root then, we are subtracting four times five times negative 32. Now, four times five times negative 32 is negative 640. So we are subtracting negative 640, which means overall we’re adding 640. But again, you need to be really careful with the negative signs here. Simplifying under the square root, 49 plus 640 is 689. So, we have that 𝑥 is equal to seven plus or minus the square root of 689 over 10.

Now, we’re asked to give our values as decimals, so we now need to use our calculator to evaluate each of these. The first root or value in our solution set is seven plus the square root of 689 all over 10, which is equal to 3.32488 continuing. The second root or second value in our solution set is seven minus the square root of 689 all over 10. And that’s equal to negative 1.92488. We’re asked to give the values to three decimal places, and in each case, the fourth decimal place is an eight. So, we round up. We can then give our two roots in set notation. The solution set to this equation is 3.325, negative 1.925.