To solve quadratic inequalities in Year 11 Math, there are some simple techniques that can help make things easier. Let’s break it down step by step!
First, you need to know what a quadratic inequality is. It usually looks like this: ( ax^2 + bx + c < 0 ) or ( ax^2 + bx + c > 0 ).
Start by looking at the quadratic function ( y = ax^2 + bx + c ). Understanding its graph will help you see where it is above or below the x-axis.
Next, solve the related quadratic equation ( ax^2 + bx + c = 0 ) to find the roots. You can use the quadratic formula to do this:
The roots are important points on the x-axis where the function changes from positive to negative or vice versa.
After finding the roots, divide the number line into intervals based on these roots.
For example, if the roots are ( r_1 ) and ( r_2 ), your intervals will look like this:
Pick a test point from each interval and plug it into your inequality. This will show you which intervals meet the condition of being greater than or less than zero.
After testing intervals, write your solution in interval notation. For instance, if your inequality was ( x^2 - 5x + 6 < 0 ) and the solution is from ( r_1 = 2 ) to ( r_2 = 3 ), you would write it as ( (2, 3) ).
Finally, drawing a graph of the quadratic function can be really helpful. It gives you a visual clue about where the function is above or below the x-axis. This can help you confirm what you found with your calculations.
By using these techniques—understanding the function, finding roots, testing intervals, and drawing the graph—you can easily solve quadratic inequalities!
To solve quadratic inequalities in Year 11 Math, there are some simple techniques that can help make things easier. Let’s break it down step by step!
First, you need to know what a quadratic inequality is. It usually looks like this: ( ax^2 + bx + c < 0 ) or ( ax^2 + bx + c > 0 ).
Start by looking at the quadratic function ( y = ax^2 + bx + c ). Understanding its graph will help you see where it is above or below the x-axis.
Next, solve the related quadratic equation ( ax^2 + bx + c = 0 ) to find the roots. You can use the quadratic formula to do this:
The roots are important points on the x-axis where the function changes from positive to negative or vice versa.
After finding the roots, divide the number line into intervals based on these roots.
For example, if the roots are ( r_1 ) and ( r_2 ), your intervals will look like this:
Pick a test point from each interval and plug it into your inequality. This will show you which intervals meet the condition of being greater than or less than zero.
After testing intervals, write your solution in interval notation. For instance, if your inequality was ( x^2 - 5x + 6 < 0 ) and the solution is from ( r_1 = 2 ) to ( r_2 = 3 ), you would write it as ( (2, 3) ).
Finally, drawing a graph of the quadratic function can be really helpful. It gives you a visual clue about where the function is above or below the x-axis. This can help you confirm what you found with your calculations.
By using these techniques—understanding the function, finding roots, testing intervals, and drawing the graph—you can easily solve quadratic inequalities!