Math Problem Statement
Solution
The system of equations given in the image is:
We can solve this system step by step.
Step 1: Solve for in the second equation
The second equation is . We can solve for in terms of :
Step 2: Substitute into the first equation
Now substitute this expression for into the first equation :
Step 3: Expand and solve the quadratic equation
Expanding the equation gives: Now, bring all terms to one side: Divide through by 2 to simplify:
Step 4: Solve the quadratic equation
This quadratic equation can be solved using the quadratic formula: Here, , , and . Plugging these values into the quadratic formula: Thus, the two solutions for are:
Step 5: Find corresponding values
Now, substitute each value into to find the corresponding :
- When :
- When :
Final Solution
The solutions to the system are:
Would you like further details or have any questions about this solution?
Here are 5 related questions for further exploration:
- How do you factor a quadratic equation like without using the quadratic formula?
- What happens if there are no real solutions for when solving a system of equations?
- How would you solve a system of equations graphically?
- What is the geometric interpretation of the solutions to a system of two equations?
- Can this system of equations be solved using matrix methods?
Tip: Always check your solutions by plugging them back into the original equations to verify they satisfy both equations.
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Math Problem Analysis
Mathematical Concepts
Algebra
Systems of Equations
Quadratic Equations
Formulas
Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Substitution method
Theorems
Quadratic Formula Theorem
Suitable Grade Level
Grades 9-10
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