Lesson video

Hi, I'm Miss Davies.

In this lesson, we're going to be using the quadratic formula to solve any quadratic equation.

The quadratic formula is used to solve quadratic equations that are in the form axe squared add bx add c equals zero.

The quadratic formula is that x is equal to negative b add or subtract the square root of b squared subtract four ac.

This is all divided by two a.

A and b are the coefficients of x squared and x.

C represents the constant.

Our first example is asking us to solve the equation four x squared subtract six x add one is equal to zero using the quadratic formula.

We have been asked to give the answer to two decimal places.

Looking at this equation, we can see that our values for a is four, b is negative six, and c is one.

We can then substitute this into the formula.

This gives x is equal to negative negative six add or subtract the square root of negative six squared subtract four multiplied by four multiplied by one.

This is all then divided by two multiplied by four.

At the start of this calculation, we are subtracting a negative number.

This is equivalent to adding.

If I simplify this calculation, it gives us six add or subtract the square root of 20 all divided by eight.

We can then use a calculator to work out these two calculations.

This gives us the solutions of x is equal to 1.

31 or x is equal to 0.

19.

Both of these are correct to two decimal places.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

If you didn't get these solutions, check the calculation that you have typed into your calculator to make sure that it is correct.

In this next example, we've been asked to solve a quadratic equation that isn't equal to zero.

We've just learned that the quadratic formula can only be used to solve equations that are equal to zero.

We're going to start off by making the equation equal to zero.

To do this, we're going to subtract five x and subtract 12 from both sides of the equation.

This gives us three x squared subtract x subtract 17 is equal to zero.

From this, we can identify our values for a, b, and c.

These are three, negative one, and negative 17.

We can then substitute these values into the quadratic formula.

This gives us that x is equal to subtract negative one add or subtract the square root of negative one squared subtract four multiplied by three multiplied by negative 17.

This is all then divided by two multiplied by three.

Because we are subtracting a negative number at the start of this calculation and this is equivalent to adding, we can rewrite that as simply one.

We can then simplify the rest of this calculation to give us that x is equal to one add or subtract the square root of 205 divided by six.

By typing these two calculations into our calculators, we can find that x is equal to 2.

55 or x is equal to negative 2.

22.

Both of these solutions are correct to two decimal places.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

You should have made these equations be equal to zero before substituting the values of a, b, and c into the quadratic formula.

In this next example, we have been told that the area of a playing field is 7,700 metres squared.

The length of the playing field is x metres and the width of the playing field is 10 metres less.

We have been asked to find the width of the playing field.

Let's start by drawing a diagram to represent the playing field.

We've been told that the area is 7,700 square metres, that the length is x metres, and the width is 10 metres less.

We can write this as x subtract 10.

We can then create a formula or an equation to represent the area.

To find the area of the playing field, we need to multiply x by x subtract 10.

This is equal to 7,700.

By expanding the brackets, we can say that x squared subtract 10 x is equal to 7,700.

To solve quadratic equation using the quadratic formula, it must be equal to zero.

To make this equation equal to zero, we're going to subtract 7,700 from both sides of the equation.

This gives us x squared subtract 10 x subtract 7,700 is equal to zero.

From this equation, we can find that our value for a is one.

Our value for b is negative 10, and our value for c is negative 7,700.

We can then substitute these three values into the quadratic formula.

Next, we can simplify this calculation down, then type the two equations into our calculator to give the solution of x is equal to 92.

9 or x is equal to negative 82.

9.

Both of these are correct to three significant figures, as is stated in the question.

Since the length x is a measurement, x must be 92.

9.

The width is 10 less than x.

This means that the width is 82.

9 metres, correct to three significant figures.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

As the length BC is the hypotenuse of the right-angled triangle, we can apply Pythagoras's theorem to the lengths.

In our next example, Sarah is solving a quadratic equation using the formula.

This is her working out.

We have been asked to find the quadratic equation that Sarah is solving, giving our answer in the form axe squared add bx add c is equal to zero.

I've written down the quadratic formula so that we can refer to it as we're going through our calculations.

We can see that the denominator of Sarah's working out is 14.

This means that two a equals 14, so a is equal to seven.

Next, we can see that negative b is 10, so b must be equal to negative 10.

We know that b squared subtract four ac is 268.

This means that 100 subtract 28 c is equal to 268.

I have used the values that we've already found for a and b to form this equation.

We can then solve this equation to find that six is equal to c, or c is equal to six.

This tells us that the equation that Sarah is solving is seven x squared subtract 10 x subtract six is equal to zero.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

Matt's equation is nine x squared subtract 11 x subtract one is equal to zero.

That's all for this lesson.

Thanks for watching.