Question

Solve \(x^{2}=3-x\).

Short answer

x = -3, 1

Step-by-step solution

Rewrite the Equation

Move all terms to one side of the equation to set the equation to zero: \[ x^2 + x - 3 = 0 \]

Factor the Quadratic Equation

Factorize the quadratic equation \( x^2 + x - 3 \). We look for two numbers that multiply to -3 and add up to 1.These numbers are 3 and -1. Therefore:\[ x^2 + x - 3 = (x + 3)(x - 1) = 0 \]

Solve for x

Set each factor equal to zero and solve for \( x \):For \( x + 3 = 0 \):\[ x = -3 \]For \( x - 1 = 0 \):\[ x = 1 \]

Key concepts

quadratic equations

Quadratic equations are equations of the form \[ ax^2 + bx + c = 0 \], where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable.
The highest exponent of the variable is 2, which gives it the name 'quadratic'.
Solving quadratic equations is a fundamental skill in algebra and is crucial for higher-level math concepts.
Common methods to solve these include:

• Factoring
• Using the quadratic formula
• Completing the square

Understanding how to manipulate and solve these types of equations will be beneficial in various mathematical and practical applications.

factoring

Factoring is one of the most common methods to solve quadratic equations.
It involves breaking down a quadratic equation into the product of its simpler binomial factors.
In the given problem, \[ x^2 + x - 3 = 0 \], we aim to express it as a product of binomials.
To factor a quadratic equation, we look for two numbers that multiply to the constant term (in this case, -3) and add up to the linear coefficient (in this case, 1).
In this case, the numbers 3 and -1 work because \[ 3 \times -1 = -3 \] and \[ 3 + (-1) = 1 \].
So, we can write the equation as: \[ (x + 3)(x - 1) = 0 \] .
This step simplifies the equation, making it easier to find the solutions for \(x\).

solving equations

The final step in solving quadratic equations by factoring is to set each binomial factor equal to zero and solve for the variable.
This is based on the 'Zero Product Property', which states that if the product of two factors is zero, then at least one of the factors must be zero.
For the factored equation \[ (x + 3)(x - 1) = 0 \] we solve:
\( x + 3 = 0 \) which gives \[ x = -3 \] and
\( x - 1 = 0 \) which gives \[ x = 1 \].
So, the solutions to the original quadratic equation are \(x = -3\) and \(x = 1\).
These solutions can be substituted back into the original equation to verify their correctness.
In summary, solving quadratic equations involves:

• Rewriting the equation
• Factoring
• Using the solutions to find the values of \(x\)

Applying these steps methodically helps break down complex problems into manageable tasks.