Video: Pack 1 β€’ Paper 1 β€’ Question 10

Pack 1 β€’ Paper 1 β€’ Question 10

04:01

Video Transcript

Solve π‘₯ squared plus eight π‘₯ plus 11 is equal to zero. Write your answer in the form π‘Ž plus or minus root 𝑏, where π‘Ž and 𝑏 are integers.

Usually, we try to solve a quadratic equation by factorizing first. However, the question has shown us that our answer will be in surd form. This means it will need to be solved either using the quadratic formula or by completing the square. Now, we can’t use either method. However, completing the square does involve fewer steps and therefore there are fewer opportunities to make a mistake.

Let’s look at the process for completing the square. A quadratic equation in completing the square form would look like this: π‘₯ plus π‘Ž all squared plus 𝑏. Now, π‘Ž and 𝑏 don’t necessarily need to be integers and it’s important to note that they are not the same letters as the ones in the question. We have used π‘Ž and 𝑏 though because these are the letters that are most commonly used when describing completed square form.

First, let’s find the value of π‘Ž which is the number inside the bracket. We do this by halving the coefficient of π‘₯. In this equation, the coefficient of π‘₯ is eight. So halving it gives us four. That gives us π‘₯ plus four all squared as the first part of our completed square form. Now, let’s look at why we actually do this.

Imagine we were going to expand π‘₯ plus four all squared back out again. Remember to square something actually means to multiply it by itself. So π‘₯ plus four squared is the same as π‘₯ plus four multiplied by π‘₯ plus four. We can use the FOIL method to multiply two brackets. π‘₯ multiplied by π‘₯ is π‘₯ squared, π‘₯ multiplied by four is four π‘₯, and four multiplied by π‘₯ is also four π‘₯. These first three terms do actually give us π‘₯ squared plus eight π‘₯ which is what we wanted from our question. However, when we multiply the final part of each bracket, that’s four multiplied by four, we get 16 not 11.

To counteract this then, we subtract 16 from our bracketed expression. Then, we replace π‘₯ squared plus eight π‘₯ with π‘₯ plus four all squared minus 16 in our original equation. Negative 16 plus 11 is negative five. So our completed square form is π‘₯ plus four squared minus five.

The question is asking us though to solve π‘₯ squared plus eight π‘₯ plus 11 is equal to zero. We can now replace π‘₯ squared plus eight π‘₯ plus 11 with our completed square form. Don’t be tempted to reexpand this bracket out. Instead, we’ll solve this equation by applying the relevant inverse operations.

We can start by adding five to both sides. That gives us π‘₯ plus four squared is equal to five. Next, we can find the square root of both sides. Remember a square root will always have two solutions: a positive and a negative. So we can add this little symbol which means plus and minus to show that we’ve got two separate answers. Finally, we’ll subtract four from both sides. π‘₯ is, therefore, equal to negative four plus or minus the square root of five.

We can compare this to the form given in the question which was π‘Ž plus or minus the square root of 𝑏. We haven’t been asked to identify what π‘Ž and 𝑏 are. But we can see by comparing them to our equation that π‘Ž is negative four and 𝑏 is five.

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