7.2 The Quadratic Formula & The Discriminant

The Quadratic Formula & The Discriminant

Quick Summary: When a quadratic equation cannot be factored easily with integers, the Quadratic Formula guarantees a universal method to find all solutions. The formula is derived directly from completing the square on the standard form $ax^2 + bx + c = 0$. The expression under the radical, $b^2 - 4ac$, is known as the discriminant ($D$), and it dictates whether the equation has two real solutions, one repeated real solution, or no real solutions.

On the HiSET Mathematics subtest, the Quadratic Formula is not on the official formula sheet — PSI names it explicitly as a formula you must know before test day, so commit it to memory. However, understanding how to substitute values accurately, manage negative signs, and interpret the discriminant algebraically and graphically is essential for a high score.

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The Quadratic Formula Anatomy

For any quadratic equation in standard form $ax^2 + bx + c = 0$ where $a \neq 0$:

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

                      Anatomy of the Quadratic Formula
                                Opposite of b
                                     │
                                     ▼
                                   -b  ±  √(b² - 4ac) ◄── Discriminant (D)
                              x = ────────────────────
                                          2a ◄── Double the leading coefficient
                                           │
                        Entire numerator is divided by 2a

Step-by-Step Evaluation Protocol

1. Ensure Standard Form: Move all terms to one side so the equation equals zero: $ax^2 + bx + c = 0$.
2. Identify Coefficients with Signs: Write down explicit values for $a$, $b$, and $c$. Include negative signs!
3. Calculate the Discriminant First: Evaluate $D = b^2 - 4ac$ inside parentheses: $(b)^2 - 4(a)(c)$.
4. Substitute into Formula: Replace $-b$ with $-1 \cdot b$ and denominator with $2 \cdot a$.
5. Simplify the Radical and Fraction: Simplify $\sqrt{D}$ and reduce the fractional coefficients by factoring out common divisors from the entire numerator.

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Worked Example 1: Quadratic Formula with Irrational Roots

Solve: $2x^2 - 6x + 1 = 0$

Step 1: Identify coefficients:

$$a = 2, \quad b = -6, \quad c = 1$$

Step 2: Calculate the discriminant ($D = b^2 - 4ac$):

$$D = (-6)^2 - 4(2)(1) = 36 - 8 = 28$$

Step 3: Substitute into the Quadratic Formula:

$$x = \frac{-(-6) \pm \sqrt{28}}{2(2)} = \frac{6 \pm \sqrt{28}}{4}$$

Step 4: Simplify the radical $\sqrt{28}$:

$$\sqrt{28} = \sqrt{4 \cdot 7} = 2\sqrt{7}$$ $$x = \frac{6 \pm 2\sqrt{7}}{4}$$

Step 5: Factor out common factor from the numerator and reduce:

$$x = \frac{2(3 \pm \sqrt{7})}{4} = \frac{3 \pm \sqrt{7}}{2}$$

The two exact solutions are $x = \frac{3 + \sqrt{7}}{2}$ and $x = \frac{3 - \sqrt{7}}{2}$.

The Discriminant Deep-Dive ($D = b^2 - 4ac$)

The discriminant determines the nature and quantity of solutions without requiring complete formula evaluation.

Comprehensive Discriminant Reference Table

               Visualizing the Discriminant and x-Intercepts

         D > 0 (Two Intercepts)     D = 0 (One Tangent)       D < 0 (Zero Intercepts)
            y                         y                         y
            │      /\                 │      /\                 │      /\   (Above axis)
            │     /  \                │     /  \                │     /  \
       ─────┼────•────•──── x    ─────┼────•─────── x    ─────┼────────────── x
            │   x₁    x₂              │  Vertex (x₁=x₂)
            │                         │                         │

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Strategy: Choosing the Most Efficient Solving Method

On a timed standardized exam like the HiSET, picking the right method saves critical minutes:

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High-Frequency Quadratic Formula Traps

1. The Parentheses Square Trap: When $b = -5$, calculating $b^2$ as $-5^2 = -25$ is incorrect. Squaring a negative number always yields a positive result: $(-5)^2 = +25$.
2. The Double Negative Sign Trap: If $b = -7$, then $-b = -(-7) = +7$. Forgetting to flip the sign is the most common HiSET error.
3. The Partial Division Error: In $\frac{6 \pm 2\sqrt{5}}{2}$, you must divide both terms in the numerator by $2$: $\frac{6}{2} \pm \frac{2\sqrt{5}}{2} = 3 \pm \sqrt{5}$. Canceling only the $6$ to write $3 \pm 2\sqrt{5}$ is invalid algebra.