Sure! Here’s a simpler way to explain how to solve a linear-quadratic system of equations, like a 6th grader might do it.
What is a Linear-Quadratic System?
A linear-quadratic system has one straight line equation and one curved (quadratic) equation. For example, let’s look at these two equations:
- Linear equation: \( y = 2x + 3 \) (this is a straight line)
- Quadratic equation: \( y = x^2 - 4 \) (this makes a curve, like a U-shape)
Step 1: Set the Equations Equal
Both equations equal \( y \), so we can set them equal to each other:
\[ 2x + 3 = x^2 - 4 \]
Step 2: Rearrange the Equation
Now, let’s rearrange the equation to get everything on one side. We want \( 0 \) on the other side:
\[ 0 = x^2 - 2x - 4 - 3 \]
This then simplifies to:
\[ x^2 - 2x - 7 = 0 \]
Step 3: Solve the Quadratic Equation
Next, we can use the quadratic formula to find out what \( x \) values work. The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case:
- \( a = 1 \) (from \( x^2 \))
- \( b = -2 \) (from \(-2x\))
- \( c = -7 \)
Let's plug in these numbers:
\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1} \] \[ x = \frac{2 \pm \sqrt{4 + 28}}{2} \] \[ x = \frac{2 \pm \sqrt{32}}{2} \] \[ x = \frac{2 \pm 4\sqrt{2}}{2} \] \[ x = 1 \pm 2\sqrt{2} \]
So we get two \( x \) values:
- \( x = 1 + 2\sqrt{2} \)
- \( x = 1 - 2\sqrt{2} \)
Step 4: Find the \( y \) Values
Now, let’s plug these \( x \) values back into the linear equation \( y = 2x + 3 \) to find \( y \):
For \( x = 1 + 2\sqrt{2} \):
\[ y = 2(1 + 2\sqrt{2}) + 3 \] \[ y = 2 + 4\sqrt{2} + 3 \] \[ y = 5 + 4\sqrt{2} \]
For \( x = 1 - 2\sqrt{2} \):
\[ y = 2(1 - 2\sqrt{2}) + 3 \] \[ y = 2 - 4\sqrt{2} + 3 \] \[ y = 5 - 4\sqrt{2} \]
Step 5: Write the Final Answers
So, we found two pairs of points where the line and the curve meet:
- First point: \( \left(1 + 2\sqrt{2}, 5 + 4\sqrt{2}\right) \)
- Second point: \( \left(1 - 2\sqrt{2}, 5 - 4\sqrt{2}\right) \)
These points are where the straight line and the U-shaped curve cross each other. And that's how you solve a linear-quadratic system!