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Find the Second Solution Set for the Simultaneous Equations $x^2+3y=4$ and $y=x-2$
Mathematics
Grade 10 of Junior High School
Question Content
There are two sets of solutions to the simultaneous equations: $x^2+3y = 4$ and $y = x-2$. One solution set is $x=2, y=0$. Find the other solution set.
Correct Answer
x = -1, y = -3
Detailed Solution Steps
1
Step 1: Substitute the linear equation $y = x-2$ into the quadratic equation $x^2+3y = 4$. Replace $y$ in the quadratic equation with $x-2$, getting $x^2 + 3(x-2) = 4$.
2
Step 2: Expand and simplify the equation. Distribute the 3: $x^2 + 3x - 6 = 4$. Then rearrange it into standard quadratic form by subtracting 4 from both sides: $x^2 + 3x - 10 = 0$.
3
Step 3: Factor the quadratic equation. We need two numbers that multiply to -10 and add to 3, which are 5 and -2. So the equation factors to $(x+5)(x-2) = 0$.
4
Step 4: Solve for $x$. Set each factor equal to 0: $x+5=0$ gives $x=-1$, and $x-2=0$ gives $x=2$ (this is the given solution).
5
Step 5: Find the corresponding $y$-value for $x=-1$. Substitute $x=-1$ into $y = x-2$, so $y = -1 - 2 = -3$.
Knowledge Points Involved
1
Solving Simultaneous Equations (Substitution Method)
This method is used to solve a system of equations where one equation is linear and the other is non-linear. You substitute the expression for one variable from the linear equation into the non-linear equation to get an equation with a single variable, which can then be solved. It is widely used for systems combining linear and quadratic equations.
2
Quadratic Equation in Standard Form
A quadratic equation in standard form is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a \\neq 0$. This form is necessary for factoring, using the quadratic formula, or completing the square to find the roots of the equation.
3
Factoring Quadratic Equations
Factoring a quadratic equation involves rewriting it as a product of two linear binomials. For an equation $x^2 + bx + c = 0$, you find two numbers that multiply to $c$ and add to $b$, then write the equation as $(x+m)(x+n)=0$ where $m$ and $n$ are those two numbers. This allows you to solve for $x$ using the zero product property.
4
Zero Product Property
The zero product property states that if $ab = 0$, then either $a = 0$, $b = 0$, or both $a$ and $b$ are 0. This property is used to solve factored quadratic equations by setting each factor equal to zero and solving for the variable.
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