Question

Choose a method to solve the quadratic equation. Explain your choice. $$x^{2}-x-20=0$$

Short answer

The solutions to the equation \(x^{2}-x-20=0\) are \(x=5\) and \(x=-4\).

Step-by-step solution

Factor the quadratic equation

To factor the quadratic equation, we look for two numbers that multiply to -20 (the constant term) and add to -1 (the coefficient of x). Those numbers are -5 and 4 because -5 times 4 equals -20 and -5 plus 4 equals -1. Thus, our equation can be expressed as: \(x^{2}-x-20=(x-5)(x+4)=0\)

Use Zero-Product Property

The next step is to use the zero-product property, which states that if a product of factors equals zero, then at least one of the factors must be zero. Therefore, we set each factor equals to zero and solve for x: \(x-5=0\), which leads to \(x=5\) and \(x+4=0\), leading to \(x=-4\).

Check the solutions

Always check the found solutions. Substitute \(x=5\) and \(x=-4\) into original equation \(x^{2}-x-20=0\). After calculation, for both x-values, we get the equality 0=0, which confirms our solution.