3.3 Quadratic Equations & Quadratic Inequalities

3.3 Quadratic Equations & Quadratic Inequalities

A quadratic equation is a second-degree polynomial equation in one variable. The standard form of a quadratic equation is:

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

where $a, b,$ and $c$ are real numbers with $a \neq 0$. The exponent $2$ makes this a non-linear equation whose graphs form parabolas.

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1. The Four Core Algebraic Solving Methods

Depending on the algebraic structure of a quadratic equation, four distinct methods can be employed.

Method 1: Factoring and Zero-Product Property

If a quadratic expression $ax^2 + bx + c$ can be factored into linear binomials $(px + q)(rx + s) = 0$, the Zero-Product Property states that at least one factor must equal zero:

$$px + q = 0 \quad \text{or} \quad rx + s = 0$$

Worked Example 1: Factoring

Solve $3x^2 - 7x - 6 = 0$.

1. Find two numbers that multiply to $a \cdot c = 3(-6) = -18$ and add to $b = -7$. Those numbers are $-9$ and $+2$.
2. Rewrite the middle term: $3x^2 - 9x + 2x - 6 = 0$.
3. Factor by grouping: $3x(x - 3) + 2(x - 3) = (3x + 2)(x - 3) = 0$.
4. Set factors to zero: $$3x + 2 = 0 \implies x = -\frac{2}{3}$$ $$x - 3 = 0 \implies x = 3$$

Solution set: $\left\{-\frac{2}{3}, 3\right\}$.

Method 2: The Square Root Property

If an equation can be written in the form $u^2 = k$ (where $u$ is an algebraic expression and $k \ge 0$), then:

$$u = \pm \sqrt{k}$$

Worked Example 2: Square Root Property

Solve $(2x - 5)^2 = 12$.

1. Apply square root to both sides: $$2x - 5 = \pm \sqrt{12} = \pm 2\sqrt{3}$$
2. Isolate $x$: $$2x = 5 \pm 2\sqrt{3} \implies x = \frac{5 \pm 2\sqrt{3}}{2}$$

Method 3: Completing the Square

Completing the square transforms any quadratic polynomial into a perfect square trinomial plus a constant.

Algorithm:

1. Rewrite $ax^2 + bx + c = 0$ as $x^2 + \frac{b}{a}x = -\frac{c}{a}$ (divide by $a$).
2. Add $\left(\frac{b}{2a}\right)^2$ to both sides.
3. Factor the left side as $\left(x + \frac{b}{2a}\right)^2$.
4. Solve using the Square Root Property.

Worked Example 3: Completing the Square

Solve $2x^2 + 8x - 10 = 0$.

1. Divide by 2: $x^2 + 4x - 5 = 0 \implies x^2 + 4x = 5$.
2. Half of coefficient 4 is 2; square it ($2^2 = 4$). Add 4 to both sides: $$x^2 + 4x + 4 = 5 + 4 \implies (x + 2)^2 = 9$$
3. Take square roots: $x + 2 = \pm 3$.
4. $x = -2 + 3 = 1$ or $x = -2 - 3 = -5$. Solution set: $\{-5, 1\}$.

Method 4: The Quadratic Formula

Derived directly by completing the square on the general equation $ax^2 + bx + c = 0$, the Quadratic Formula solves any quadratic equation:

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

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2. Discriminant Analysis ($D = b^2 - 4ac$)

The expression under the radical in the quadratic formula, $D = b^2 - 4ac$, is called the discriminant. It dictates the number and numerical character of the roots without fully completing the radical arithmetic.

Worked Example 4: Quadratic Formula with Complex Conjugate Roots

Problem: Solve $2x^2 - 4x + 5 = 0$.

Step-by-Step Solution:

1. Identify coefficients: $a = 2, b = -4, c = 5$.
2. Compute discriminant: $$D = b^2 - 4ac = (-4)^2 - 4(2)(5) = 16 - 40 = -24$$
3. Since $D = -24 < 0$, the roots are complex conjugates involving $i = \sqrt{-1}$: $$x = \frac{-(-4) \pm \sqrt{-24}}{2(2)} = \frac{4 \pm i\sqrt{24}}{4} = \frac{4 \pm 2i\sqrt{6}}{4} = 1 \pm \frac{\sqrt{6}}{2}i$$

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3. Solving Quadratic Inequalities & Sign Charts

A quadratic inequality can be written in forms such as $ax^2 + bx + c > 0$, $ax^2 + bx + c \le 0$, etc.

Step-by-Step Sign Chart Algorithm

1. Standard Form: Move all terms to the left side so that zero appears on the right side.
2. Find Critical Numbers: Solve the boundary equation $ax^2 + bx + c = 0$ to find real roots.
3. Partition Test Intervals: Plot critical numbers on a real number line to divide it into open test intervals.
4. Evaluate Interval Signs: Select a test value $x_t$ within each open interval and substitute it into the quadratic polynomial to record its sign ($+$ or $-$).
5. State Solution: Select intervals that satisfy the inequality operator ($>$ or $\ge$ requires $+$; $<$ or $\le$ requires $-$). Include endpoints for $\le, \ge$ unless restricted.

Worked Example 5: Quadratic Inequality

Problem: Solve $x^2 - 2x - 8 \ge 0$.

Step-by-Step Solution:

1. Factor: $(x - 4)(x + 2) \ge 0$.
2. Critical boundary points: $x = -2$ and $x = 4$.
3. Test intervals: $(-\infty, -2)$, $(-2, 4)$, and $(4, \infty)$.
4. Test point evaluation:
   • Interval $(-\infty, -2)$, test $x = -3$: $(-3 - 4)(-3 + 2) = (-7)(-1) = +7 > 0$ (+)
   • Interval $(-2, 4)$, test $x = 0$: $(0 - 4)(0 + 2) = (-4)(2) = -8 < 0$ (-)
   • Interval $(4, \infty)$, test $x = 5$: $(5 - 4)(5 + 2) = (1)(7) = +7 > 0$ (+)
5. Inequality calls for $\ge 0$ (positive or zero). Endpoints $-2$ and $4$ are included. Solution set: $(-\infty, -2] \cup [4, \infty)$.

Worked Example 6: Strict Quadratic Inequality

Problem: Solve $2x^2 - 5x - 3 < 0$.

Step-by-Step Solution:

1. Factor: $(2x + 1)(x - 3) < 0$.
2. Critical points: $x = -\frac{1}{2}$ and $x = 3$.
3. Test intervals: $\left(-\infty, -\frac{1}{2}\right)$, $\left(-\frac{1}{2}, 3\right)$, and $(3, \infty)$.
4. Evaluating test points reveals negative output only on $\left(-\frac{1}{2}, 3\right)$.
5. Since the inequality is strictly $<$, endpoints are excluded. Solution set: $\left(-\frac{1}{2}, 3\right)$.