If you’re struggling with this week’s questions, this walkthrough should help you get the skills you need.

This week, we’re exploring:

- Solving Quadratics by Factorising
- Reflection
- Multiplying Mixed Numbers
- Circle Theorems
- Angles in Parallel Lines

### Question 1: Solving Quadratics by Factorising

Factorising is **one way** to solve quadratic equations. It doesn’t work for all quadratics but, if the question is on a non-calculator paper, the chances are you can solve the quadratic by factorising it.

**Solve \(6x^2 – 19x + 15 = 0\). **

First, we factorise the left-hand side. You may want to look back at week 5 if you think you need to revisit this.

To factorise, we look for a pair of numbers that **multiply **together to give 𝑎𝑐 = 90 and **add **together to give -19. The only pair of numbers that fits both requirements is -9 and -10. We use this to split our middle term and then factorise in pairs:

\(\begin{aligned}6x^2-9x-10x+15 &= 3x(2x-3)-5(2x-3)\\ &= (3x-5)(2x-3) \end{aligned}\)

So, \((3x-5)(2x-3)=0\). For two numbers to multiply together to give 0, one of them must be zero.

\(3x-5=0\)

\(x=\frac{5}{3}\)

or

\(2x-3=0\)

\(x=\frac{3}{2}\)

So, the **two solutions** to ** \(6x^2 – 19x + 15 = 0\) **are

**and**

**\(x=\frac{5}{3}\)****.**

**\(x=\frac{3}{2}\)**### Question 2: Reflection

When reflecting a shape, you will be given a straight line to reflect a shape in. Usually, you will need to draw this line in first.

**Reflect the shape below in the line with equation 𝑦 = 𝑥.**

First, we need to draw the line with equation 𝑦 = 𝑥. This will go through the points (0, 0), (1, 1), (2, 2) and so on.

Now, we think about how the reflection will affect each vertex (corner) of the shape. If we consider the perpendicular distance to the line, the reflected vertex will be the **same distance** on the other side of the line.

Joining these three vertices together gives us our final shape:

**Describe the single transformation that maps shape A onto shape B.**

The key here is to find the straight line that is the same distance from shape A and from shape B.

This is the line with equation **𝑥 = 5**.

### Question 3: Multiplying Mixed Numbers

When multiplying fractions, we simply multiply the **numerators **(the top numbers) and then multiply the **denominators **(the bottom numbers). Then, if we need to, we simplify our answer. With mixed numbers, we need to change them to **improper fractions first**.

**Let’s say we want to calculate** \( 2\frac{1}{7}\times 1\frac{5}{6}\)**.**

\( \begin{aligned} 2\frac{1}{7}\times 1\frac{5}{6} &= \frac{15}{7}\times \frac{11}{6} \\ &= \frac{15 \times 11}{7 \times 6} \\ &= \frac{165}{42} \end{aligned}\)

We need to see if we can simplify our answer by checking for any **common factors** of 165 and 42. In this case, they are both divisible by 3 so we divide both the numerator and denominator by 3:

\( \begin{aligned} \frac{165}{42} = \frac{55}{14}\end{aligned}\)

55 and 14 have no common factors so this fraction is in its simplest form. Now, we just put it back as a mixed number.

\( \begin{aligned} \frac{55}{14} = 3\frac{13}{14}\end{aligned}\)

### Question 4: Circle Theorems

These are the two circle theorems that are relevant this week.

Given that O is the centre of the circle and PQ is a tangent to the circle, find the size of the angle marked 𝑥.

Alternate segment theorem means that angle BCQ = angle BAC so angle BCQ = 28°.

OC is a radius and CQ is a tangent, so angle OCQ = 90°.

Therefore:

\( \begin{aligned} x &= 90-28 \\ &= 62° \end{aligned} \)

### Question 5: Angles in Parallel Lines

There are three rules you need to know for angles in parallel lines:

You might also need to use these rules – they are often used in questions that include angles in parallel lines.

**Let’s find the labelled missing angles in the shape below.**

The best way to start is to write down any angles you can see. For example, you might spot that 83° and are on a **straight line**. Label on the diagram and then write out your working **and **the reason.

= 180 – 83

= 97° *Angles on a straight line add up to 180°.*

Write as much detail as you can in the reason – most exam boards have keywords they are looking for, so it is worth looking at mark schemes to see what these are.

Next, we might spot that and 37° are **corresponding angles**. Corresponding angles make an F shape on the diagram.

= 37° *Corresponding angles are equal*.

Finally, you could spot that and 83° are **alternate angles**. Alternate angles make a Z shape on the diagram.

= 83° *Alternate angles are equal*.

You could also have spotted that and are **co-interior angles** which add up to 180°. Co-interior angles make a C shape on the diagram.

There is always more than one way to answer a question like this. The reasons are worth a large proportion of the marks so if you can see more than one solution, you should use the one that you can explain better.

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