91Ó°ÊÓ

The substitution method is a powerful technique used to solve equations by replacing one variable with a given value. In this exercise, you are given a quadratic equation, and a specific point \((a, 4)\) on its graph. This point tells you that when \((x = a)\), the value of \((y)\) is 4. This leads to substituting 4 for \((y)\) and \((a)\) for \((x)\) in the quadratic equation. By substituting these values, you transform the original equation \((y = x^2 + 3x)\) into \((4 = a^2 + 3a)\). This step is crucial as it now sets the stage to solve for \((a)\) in the next steps.

The quadratic formula is a fundamental tool in algebra, used to solve any quadratic equation of the form \((ax^2 + bx + c = 0)\). The formula is expressed as: \[ x = \frac{{-b \, \, pm \, \, sqrt{{b^2 - 4ac}}}}{{2a}} \] In the earlier step, we transformed our substituted equation into the standard quadratic form: \((a^2 + 3a - 4 = 0)\). Here, the coefficients are \((a = 1)\), \((b = 3)\), and \((c = -4)\). Plugging these values into the quadratic formula allows us to find the possible values of \((a)\). Simplified, this delivers two potential solutions, \((a = 1)\) and \((a = -4)\). It demonstrates the flexibility and robustness of the quadratic formula in providing solutions to polynomial equations.

The discriminant is a part of the quadratic formula found under the square root, and it provides valuable information about the nature of the roots of a quadratic equation. It is calculated as \((b^2 - 4ac)\). In this exercise, after identifying \((a = 1)\), \((b = 3)\), and \((c = -4)\), the discriminant becomes \((3^2 - 4 \, \times 1 \, \times -4 = 9 + 16 = 25)\). A discriminant value that’s positive (as in this case) indicates two real and distinct roots exist. When we proceed with the quadratic formula using this discriminant, it confirms that \((a = 1)\) and \((a = -4)\) are indeed valid solutions. The discriminant succinctly guides us on the type and number of solutions to expect before fully solving the equation.