Quadratic formula Knowledge-Based Study

Study notes

1. Understanding the Formula

   Every quadratic equation can be written in standard form as:

   $$ax^2 + bx + c = 0$$

   where a, b, and c are real constants and a ≠ 0.

   When factoring is difficult or impossible, the quadratic formula provides an algebraic method to find all solutions directly from the coefficients:

   $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

   The symbol ± indicates that there are typically two potential solutions:

   • One obtained by adding the radical: $x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
   • One obtained by subtracting the radical: $x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$

   Analyzing the Discriminant

   The expression beneath the radical sign is called the discriminant, commonly denoted by the Greek letter delta ($\Delta$):

   $$\Delta = b^2 - 4ac$$

   The value of $\Delta$ determines the number and type of roots without needing to calculate the full solution:

   • If $\Delta > 0$: The equation has two distinct real solutions. If $\Delta$ is a perfect square, the roots are rational; otherwise, they are irrational.
   • If $\Delta = 0$: The equation has exactly one real solution (a repeated or double root) given by $x = -\frac{b}{2a}$.
   • If $\Delta < 0$: The equation has no real solutions (it produces two complex conjugate solutions).

   Worked Example 1

   Solve the quadratic equation:

   $$2x^2 - 5x - 3 = 0$$Step 1: Identify coefficients

   Ensure the equation is in standard form $ax^2 + bx + c = 0$:

   $a = 2$, $b = -5$, and $c = -3$.

   Step 2: Calculate the discriminant$$\Delta = b^2 - 4ac = (-5)^2 - 4(2)(-3) = 25 + 24 = 49$$

   Since $\Delta = 49 > 0$, there are two distinct real solutions.

   Step 3: Substitute into the quadratic formula$$x = \frac{-(-5) \pm \sqrt{49}}{2(2)} = \frac{5 \pm 7}{4}$$Step 4: Compute each solution

   • $x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3$
   • $x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}$

   The solution set is $\{ -\frac{1}{2}, 3 \}$.

   Worked Example 2

   Solve the quadratic equation:

   $$x^2 - 6x + 7 = 0$$Step 1: Identify coefficients

   $a = 1$, $b = -6$, $c = 7$.

   Step 2: Evaluate the discriminant$$\Delta = (-6)^2 - 4(1)(7) = 36 - 28 = 8$$Step 3: Apply the formula and simplify radicals$$x = \frac{-(-6) \pm \sqrt{8}}{2(1)} = \frac{6 \pm 2\sqrt{2}}{2}$$

   Factor out the common factor of 2 from the numerator:

   $$x = \frac{2(3 \pm \sqrt{2})}{2} = 3 \pm \sqrt{2}$$

   The exact real solutions are $x = 3 + \sqrt{2}$ and $x = 3 - \sqrt{2}$.