How can we solve the system of equations represented by y = x² + 4 and y = 2x + 1, and can you show step-by-step work?

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Solving the System of Equations

To solve the system of equations:

  • Equation 1: y = x² + 4
  • Equation 2: y = 2x + 1

We will set the two equations equal to each other since they both represent y.

Step 1: Set the equations equal to each other

From Equation 1 and Equation 2, we have:

x² + 4 = 2x + 1

Step 2: Rearranging the equation

To simplify, we’ll move all terms to one side:

x² - 2x + 4 - 1 = 0
x² - 2x + 3 = 0

Step 3: Apply the Quadratic Formula

This equation is in the standard form ax² + bx + c = 0, where:

  • a = 1
  • b = -2
  • c = 3

The Quadratic Formula is:

x = (-b ± √(b² - 4ac)) / 2a

Substituting the values of a, b, and c into the formula:

x = (2 ± √((-2)² - 4(1)(3))) / (2 * 1)
x = (2 ± √(4 - 12)) / 2
x = (2 ± √(-8)) / 2

Since we have a negative value under the square root (√(-8)), the solutions for x will be complex numbers. Calculating further, we find:

x = (2 ± 2i√2) / 2
x = 1 ± i√2

Step 4: Find y values

Now that we have determined x values, we can substitute back into either Equation 1 or Equation 2 to find corresponding y values. We’ll use Equation 2 for this:

y = 2(1 ± i√2) + 1
y = 2 ± 2i√2 + 1
y = 3 ± 2i√2

Final Solutions

The solutions to the system of equations are:

  • x = 1 + i√2, y = 3 + 2i√2
  • x = 1 – i√2, y = 3 – 2i√2

In conclusion, the system of equations has complex solutions.

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