Question

Solve the equation. Tell which solution method you used. \(x^{2}-3 x-4=0\)

Short answer

The solutions to the equation \(x^{2}-3x-4=0\) are \(x = 4\) and \(x = -1\). The factorization method was used to solve it.

Step-by-step solution

Setting the equation to zero

The equation is already set to zero. So no changes are needed.

Factorizing the equation

To factorize \(x^{2}-3x-4=0\), you need to find two numbers that multiply to \(-4\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). Those numbers are \(-4\) and \(1\). Therefore, the factored form of the equation is \((x - 4)(x + 1) = 0\)

Setting each factor equal to zero

By the Zero Factor Property, a product of factors equals zero if and only if at least one of the factors is zero. Set each factor equal to zero and solve for \(x\): \(x - 4 = 0 \Rightarrow x = 4\) and \(x + 1 = 0 \Rightarrow x = -1\)

Key concepts

Factoring Quadratics

Factoring quadratics is one of the essential techniques used in solving quadratic equations. A quadratic equation is any equation that can be written in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The goal of factoring is to break down the equation into simpler parts, specifically into a product of two binomials. To factor a quadratic equation like \(x^2-3x-4=0\), look for two numbers that multiply to the constant term (in this case, -4) and add up to the coefficient of the linear term (in this case, -3).

Once the two numbers are found, rewrite the quadratic as the product of two binomials. In our exercise, the numbers -4 and 1 fit the criteria, factorizing the equation to \(x-4)\times(x+1)=0\). This sets the stage for the next step in the solution process, utilizing the zero product property.

Zero Product Property

The zero product property is a critical rule in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. This property allows us to take an equation, such as the factored quadratic \(x-4)\times(x+1)=0\), and break it into two separate equations. Each factor is set equal to zero: \(x-4 = 0\) and \(x+1 = 0\).

The zero product property simplifies the process of finding solutions to the quadratic equation. By solving these simpler equations, we find that \(x=4\) and \(x=-1\). These values of \(x\) are the roots or solutions to the original quadratic equation. Understanding this property is essential for students as it is frequently used in algebra to solve a variety of problems.

Quadratic Formula

The quadratic formula offers an alternate way to solve quadratic equations, which is especially helpful when factoring is challenging or not possible. The formula is given by \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients of the terms in a quadratic equation \(ax^2 + bx + c = 0\).

To use the quadratic formula, you substitute the values of \(a\), \(b\), and \(c\) from your equation into the formula and simplify to find the values of \(x\) that solve the equation. Although in our exercise, factoring was the simpler method, the quadratic formula is a powerful tool as it works for any quadratic equation, no matter how complex. Using this formula ensures that you can always find the solution to a quadratic, making it an invaluable method in a student's mathematical toolkit.