Solve for x 3x^2-28=2x^2-3x
The problem presented is a quadratic equation where you are asked to find the variable "x." It involves two expressions set equal to each other with different powers of x – one side of the equation has a term with x squared, and the other side also has a term with x squared and a term with x to the first power. To solve for x, you would typically aim to simplify and rearrange the equation, hopefully leading to a simpler quadratic equation that can be solved either by factoring, completing the square, or using the quadratic formula. The goal is to isolate x and determine all possible values that satisfy the equation.
$3 x^{2} - 28 = 2 x^{2} - 3 x$
Answer
Solution:
Step 1:
Reposition the equation to place $x$ on the left-hand side. $2x^2 - 3x = 3x^2 - 28$
Step 2:
Consolidate all $x$ terms on one side of the equation.
Step 2.1:
Subtract $3x^2$ from each side. $2x^2 - 3x - 3x^2 = -28$
Step 2.2:
Combine like terms. $-x^2 - 3x = -28$
Step 3:
Balance the equation by adding $28$ to both sides. $-x^2 - 3x + 28 = 0$
Step 4:
Decompose the quadratic expression on the left.
Step 4.1:
Extract the common factor of $-1$.
Step 4.1.1:
Factor out $-1$ from $-x^2$. $-(x^2) - 3x + 28 = 0$
Step 4.1.2:
Factor out $-1$ from $-3x$. $-(x^2) - (3x) + 28 = 0$
Step 4.1.3:
Express $28$ as $-1 \cdot (-28)$. $-(x^2) - (3x) - 1 \cdot (-28) = 0$
Step 4.1.4:
Factor $-1$ from the entire expression. $-(x^2 + 3x - 28) = 0$
Step 4.2:
Factor the quadratic expression.
Step 4.2.1:
Apply the AC method to factor $x^2 + 3x - 28$.
Step 4.2.1.1:
Identify two integers whose product is $-28$ and sum is $3$. These are $-4$ and $7$.
Step 4.2.1.2:
Write the factors using these numbers. $-((x - 4)(x + 7)) = 0$
Step 5:
Recognize that if any factor equals zero, the equation is satisfied. $x - 4 = 0$ or $x + 7 = 0$
Step 6:
Solve for $x$ when $x - 4 = 0$.
Step 6.1:
Set $x - 4$ to zero. $x - 4 = 0$
Step 6.2:
Add $4$ to isolate $x$. $x = 4$
Step 7:
Solve for $x$ when $x + 7 = 0$.
Step 7.1:
Set $x + 7$ to zero. $x + 7 = 0$
Step 7.2:
Subtract $7$ to isolate $x$. $x = -7$
Step 8:
Combine the solutions to find the values that satisfy the equation. $x = 4$ or $x = -7$
Knowledge Notes:
The problem involves solving a quadratic equation by rearranging terms, combining like terms, factoring, and applying the zero-product property. Here are the relevant knowledge points:
Rearranging Equations: To solve for a variable, it's often helpful to rearrange the equation to get all terms with the variable on one side and constants on the other.
Combining Like Terms: Terms that have the same variable raised to the same power can be combined by adding or subtracting their coefficients.
Factoring: This is the process of breaking down a complex expression into simpler factors that, when multiplied together, give the original expression. Common factoring techniques include factoring out a greatest common factor and factoring trinomials.
Zero-Product Property: If the product of two factors is zero, then at least one of the factors must be zero. This property allows us to set each factor of a factored equation to zero to solve for the variable.
AC Method: A factoring technique used to factor trinomials where the coefficient of $x^2$ is not 1. It involves finding two numbers that multiply to give the product of the coefficient of $x^2$ and the constant term (AC), and add to give the coefficient of the $x$ term (B).
Solving Quadratic Equations: Once a quadratic equation is factored, each factor can be set equal to zero to find the possible values of the variable that satisfy the original equation.
