Question

Solve: \(x^{2}+3 x=4\)

Short answer

The solutions are x = -4 and x = 1.

Step-by-step solution

Move all terms to one side

Rewrite the equation by moving the constant term to the left side: \[ x^2 + 3x - 4 = 0 \]

Factor the quadratic equation

Factor the quadratic equation. We need to find two numbers that multiply to -4 (constant term) and add to 3 (coefficient of the middle term). The factors are 4 and -1. So, we can write: \[ (x + 4)(x - 1) = 0 \]

Solve for x

Set each factor equal to zero and solve for x: \[ x + 4 = 0 \] \[ x = -4 \] and \[ x - 1 = 0 \] \[ x = 1 \]

Key concepts

factoring quadratics

Quadratic equations have the general form of \[ ax^2 + bx + c = 0 \]. Factoring is one of the methods used to solve these equations. Factoring involves finding two binomials that multiply together to give you the original quadratic equation. To identify the appropriate binomials, look for two numbers that:

\[ \bullet \] Multiply to the constant term (c)
\[ \bullet \] Add up to the coefficient of the middle term (b)

For example, in the equation \[ x^2 + 3x - 4 = 0 \], we must find two numbers that multiply to -4 and add to 3. The numbers 4 and -1 satisfy these conditions: \[ 4 \times (-1) = -4 \] and \[ 4 + (-1) = 3 \]. Hence, we can write the factored form as
\[ (x + 4)(x - 1) = 0 \].

zero product property

The Zero Product Property is a fundamental concept in algebra. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. This property is key when dealing with factored quadratic equations. When a quadratic equation is factored into binomials set equal to zero, such as \[ (x + 4)(x - 1) = 0 \], we use this property to set each binomial to zero:

\[ \bullet \] \[ x + 4 = 0 \]
\[ \bullet \] \[ x - 1 = 0 \]

Solving these equations individually gives us the solutions for x. These solutions are where the original quadratic equation equals zero. In this example:

\[ \bullet \] \[ x + 4 = 0 \] gives \[ x = -4 \]
\[ \bullet \] \[ x - 1 = 0 \] gives \[ x = 1 \]

Therefore, the solutions are x = -4 and x = 1.

moving terms

In solving a quadratic equation, a crucial first step is to move all terms to one side of the equation, setting it equal to zero. This sets up the equation for easier factoring. For example, starting with \[ x^2 + 3x = 4 \], we move the constant term (4) from the right side to the left side. This is done by subtracting 4 from both sides:
\[ x^2 + 3x - 4 = 0 \]. Now, the equation is in standard quadratic form \[ ax^2 + bx + c = 0 \]. This reconfiguration is essential because factoring and other solving methods require the equation to be in this format. By moving the terms correctly, we prepare the quadratic equation for factoring, completing the square, or using the quadratic formula.