Final answer:
The factored version of the quadratic equation x² - 4x - 21 = 0 is (x + 7)(x - 3) = 0. The equation factors to these binomials because the numbers -7 and 3 multiply to give -21 and add to give -4.
Step-by-step explanation:
The equation x² - 4x - 21 = 0 is a quadratic equation that can be factored to find its roots. Factoring a quadratic equation means expressing it as a product of two binomial expressions set equal to zero. To factor the equation, we look for two numbers that multiply to give -21 (the constant term) and add to give -4 (the coefficient of the x term). The numbers -7 and 3 fulfill these conditions because (-7) × (3) = -21 and (-7) + (3) = -4. Thus, the factored form of the equation is (x + 7)(x - 3) = 0.
When the product of two factors equals zero, either one or both of the factors must be zero. This leads to two possible solutions for x: either x + 7 = 0, which gives x = -7, or x - 3 = 0, which gives x = 3. These solutions represent the x-intercepts of the parabola when graphed on a coordinate plane.