Understanding Quadratic Equations in Year 10 Math

Quadratic equations are an important topic in Year 10 Math, especially in the British school system.

A typical quadratic equation looks like this:

$$y = ax^2 + bx + c$$

Here’s what each part means:

• $a$, $b$, and $c$ are numbers called coefficients.
• $x$ is the variable we’re working with.
• We must remember that $a$ cannot be zero because, if it were, we would not have a quadratic equation anymore—it would just be a straight line.

What Are Coefficients?

In the equation $y = ax^2 + bx + c$:

• Coefficient $a$: This affects how the graph looks. If $a$ is greater than zero ($a > 0$), the graph opens upwards like a U. If $a$ is less than zero ($a < 0$), it opens downwards like an upside-down U. The bigger the number for $a$, the narrower the U will be.

• Coefficient $b$: This helps to find the location of the vertex (the highest or lowest point) of the graph on the x-axis. It’s used to calculate the line of symmetry with the formula $x = -\frac{b}{2a}$.

• Coefficient $c$: This is a constant, and it tells us where the graph meets the y-axis. This is called the y-intercept. It’s the point we get when $x$ equals zero.

Finding Roots of Quadratic Equations

The roots, or solutions, of a quadratic equation can be found in several ways, like factoring, completing the square, or using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

What is the Discriminant?

We use something called the discriminant, which is calculated as $D = b^2 - 4ac$. It helps us understand more about the roots:

• When $D > 0$: This means there are two different real roots. For instance, if we take the equation $2x^2 - 4x + 1$, the discriminant tells us:

  $$(-4)^2 - 4 \cdot 2 \cdot 1 = 16 - 8 = 8 > 0$$

  This tells us there are two different solutions.

• When $D = 0$: This means there is one real root that repeats. For example, in the equation $x^2 - 2x + 1$, we find:

  $$(-2)^2 - 4 \cdot 1 \cdot 1 = 4 - 4 = 0$$

  This shows one root, which is $x = 1$.

• When $D < 0$: There are no real roots, meaning the graph does not touch the x-axis. For example, in $x^2 + 2x + 5$, we find:

  $$2^2 - 4 \cdot 1 \cdot 5 = 4 - 20 = -16 < 0$$

  This tells us the roots are complex, or imaginary.

Summary of Key Points

1. The coefficients $a$, $b$, and $c$ shape the graph and the mathematical properties of the quadratic equation.

2. The value of $a$ decides if the graph opens up or down and how wide or narrow it is.

3. The value of $b$ helps us find the vertex and works with the axis of symmetry.

4. The value of $c$ shows where the graph crosses the y-axis.

5. The discriminant $D$ gives us clues about the roots—whether they are different, the same, or don’t exist at all.

By understanding these parts, students can learn the basics of quadratic functions. This knowledge will help them tackle more challenging math ideas, like graphing and optimization, as they continue their studies.