Solve by Factoring y=x^2+5x-6
The problem provides a quadratic equation in the form of y = x^2 + 5x - 6 and asks to find the solution(s) for the variable x by using the factoring method. Factoring involves rewriting the quadratic equation as a product of two binomials and then using the zero product property to solve for the variable x.
$y = x^{2} + 5 x - 6$
Answer
Solution:
Step 1: Rearrange the equation to set it to zero.
Step 1.1: Subtract $x^2$ from both sides to get $y - x^2 = 5x - 6$.
Step 1.2: Now subtract $5x$ from both sides to obtain $y - x^2 - 5x = -6$.
Step 1.3: Finally, add $6$ to both sides to achieve the standard quadratic form $y - x^2 - 5x + 6 = 0$.
Step 2: Isolate the variable $y$.
Step 2.1: Add $x^2$ to both sides to get $y - 5x + 6 = x^2$.
Step 2.2: Then add $5x$ to both sides to have $y + 6 = x^2 + 5x$.
Step 2.3: Subtract $6$ from both sides to isolate $y$, resulting in $y = x^2 + 5x - 6$.
Knowledge Notes:
The process of solving a quadratic equation by factoring involves several key knowledge points:
Standard Quadratic Form: A quadratic equation is typically written in the form $ax^2 + bx + c = 0$. The process involves rearranging the equation into this standard form if it is not already presented as such.
Moving Terms Across the Equals Sign: When we move terms from one side of the equation to the other, we perform the opposite operation. For example, if we move a term that is being added to one side, we subtract it from the other side, and vice versa.
Factoring Quadratics: Once the quadratic equation is in standard form, we can attempt to factor it. Factoring involves finding two binomials that multiply to give the original quadratic. For example, $(x + m)(x + n) = x^2 + (m+n)x + mn$.
Zero Product Property: If the product of two expressions is zero, then at least one of the expressions must be zero. This property allows us to set each factor equal to zero and solve for $x$.
Solving for a Variable: In the context of this problem, we are not required to solve for $x$ but rather to express $y$ in terms of $x$. This is achieved by isolating $y$ on one side of the equation.
Checking Solutions: After finding the solutions for $x$, it is good practice to substitute them back into the original equation to ensure they satisfy the equation.
In this particular problem, the goal was to rearrange the given equation to isolate $y$ and express it in terms of $x$, which is a straightforward process of moving terms across the equals sign to achieve the desired form.
