solve the simultaneous equations y2=3x2+4   y+2x=7

To solve the system of simultaneous equations:

\[
y^2 = 3x^2 + 4
\]
and
\[
y + 2x = 7
\]

### Step 1: Solve for \( y \) in terms of \( x \) from the linear equation
From the second equation, \( y + 2x = 7 \), solve for \( y \):

\[
y = 7 - 2x
\]

### Step 2: Substitute \( y = 7 - 2x \) into the first equation
Now substitute \( y = 7 - 2x \) into the first equation \( y^2 = 3x^2 + 4 \):

\[
(7 - 2x)^2 = 3x^2 + 4
\]

### Step 3: Expand and simplify
Expand \( (7 - 2x)^2 \):

\[
(7 - 2x)^2 = 49 - 28x + 4x^2
\]

So the equation becomes:

\[
49 - 28x + 4x^2 = 3x^2 + 4
\]

### Step 4: Move all terms to one side and simplify
Bring all terms to one side:

\[
49 - 28x + 4x^2 - 3x^2 - 4 = 0
\]

Simplify:

\[
x^2 - 28x + 45 = 0
\]

### Step 5: Solve the quadratic equation
Now solve the quadratic equation \( x^2 - 28x + 45 = 0 \) using the quadratic formula:

\[
x = \frac{-(-28) \pm \sqrt{(-28)^2 - 4(1)(45)}}{2(1)}
\]
\[
x = \frac{28 \pm \sqrt{784 - 180}}{2}
\]
\[
x = \frac{28 \pm \sqrt{604}}{2}
\]
\[
x = \frac{28 \pm 24.57}{2}
\]

So the two possible values for \( x \) are:

\[
x = \frac{28 + 24.57}{2} \approx 26.285
\]
or
\[
x = \frac{28 - 24.57}{2} \approx 1.715
\]

### Step 6: Solve for \( y \)
Now, substitute these values of \( x \) back into the equation \( y = 7 - 2x \):

1. For \( x \approx 26.285 \):

\[
y = 7 - 2(26.285) \approx 7 - 52.57 \approx -45.57
\]

2. For \( x \approx 1.715 \):

\[
y = 7 - 2(1.715) \approx 7 - 3.43 \approx 3.57
\]

### Final Solutions:
The two solutions for the system are:

1. \( x \approx 26.285 \), \( y \approx -45.57 \)
2. \( x \approx 1.715 \), \( y \approx 3.57 \)

Would you like to explore any specific part of the solution in more detail? Or have other questions on this? 

Here are some related questions to help further your understanding:

1. How would you solve quadratic equations using factoring?
2. What does the discriminant in the quadratic formula tell you about the number of solutions?
3. How do you verify if the solutions to the system are correct?
4. Can systems of equations have no solution or infinitely many solutions?
5. How would you solve simultaneous equations with different types (linear and quadratic)?

**Tip:** When solving quadratic equations, always check if factoring is possible before using the quadratic formula. It can save time!

Explore the step-by-step solution to solving the simultaneous equations y^2 = 3x^2 + 4 and y + 2x = 7. This solution covers the substitution method, quadratic equations, and the quadratic formula. Suitable for high school students in Grades 10-12.

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