Quadratic Equations (SAT Math)

What is a Quadratic Equation?

A quadratic equation has the form:

$ax^2 + bx + c = 0$

where $a \neq 0$ (if $a = 0$, it's just linear!)

Examples:

• $x^2 - 5x + 6 = 0$
• $2x^2 + 3x - 2 = 0$
• $x^2 = 16$ (equivalent to $x^2 - 16 = 0$)

Solving Methods

Method 1: Factoring

Best when: Quadratic factors nicely

Steps:

1. Set equation equal to zero
2. Factor the quadratic
3. Set each factor equal to zero
4. Solve for $x$

Example: Solve $x^2 - 5x + 6 = 0$

Step 1: Already set to zero ✓

Step 2: Factor
Need two numbers that multiply to 6 and add to -5: -2 and -3

$x^2 - 5x + 6 = (x - 2)(x - 3) = 0$

Step 3 & 4: Set each factor to zero
$x - 2 = 0$ → $x = 2$
$x - 3 = 0$ → $x = 3$

Solutions: $x = 2$ or $x = 3$

Method 2: Quadratic Formula

Best when: Doesn't factor nicely

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

Memorize this! It works for ANY quadratic.

Example: Solve $2x^2 + 3x - 2 = 0$

Identify: $a = 2$, $b = 3$, $c = -2$

Substitute: $x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}$

$x = \frac{-3 \pm \sqrt{9 + 16}}{4}$

$x = \frac{-3 \pm \sqrt{25}}{4}$

$x = \frac{-3 \pm 5}{4}$

Two solutions: $x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$

$x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$

Solutions: $x = \frac{1}{2}$ or $x = -2$

Method 3: Completing the Square

Best when: Equation is in form $x^2 + bx = c$

Steps:

1. Move constant to right side
2. Add $(\frac{b}{2})^2$ to both sides
3. Factor left side as perfect square
4. Take square root of both sides
5. Solve for $x$

Example: Solve $x^2 + 6x = 7$

Step 1: Already done ✓

Step 2: $(\frac{6}{2})^2 = 3^2 = 9$

$x^2 + 6x + 9 = 7 + 9$ $x^2 + 6x + 9 = 16$

Step 3: Factor left side
$( x + 3)^2 = 16$

Step 4: Take square root
$x + 3 = \pm 4$

Step 5: Solve
$x + 3 = 4$ → $x = 1$
$x + 3 = -4$ → $x = -7$

Solutions: $x = 1$ or $x = -7$

Method 4: Simple Square Roots

Best when: No $x$ term (just $ax^2 = c$)

Example: Solve $x^2 = 16$

$x = \pm\sqrt{16} = \pm 4$

Solutions: $x = 4$ or $x = -4$

DON'T FORGET THE ±! Both positive and negative square roots.

The Discriminant

The discriminant tells you about the solutions WITHOUT solving:

$\text{Discriminant} = b^2 - 4ac$

If discriminant is:

• Positive: 2 real solutions (different numbers)
• Zero: 1 real solution (repeated/double root)
• Negative: No real solutions (complex numbers)

Example: How many real solutions does $x^2 - 4x + 4 = 0$ have?

$b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$

Answer: ONE solution (it's a perfect square: $(x-2)^2 = 0$ → $x = 2$)

Parabolas and Graphs

Standard Form: $y = ax^2 + bx + c$

Key features:

Vertex (turning point):

• x-coordinate: $x = -\frac{b}{2a}$
• Plug this back in to find y-coordinate

Direction:

• If $a > 0$: Opens UP (∪ shape)
• If $a < 0$: Opens DOWN (∩ shape)

y-intercept:

• Set $x = 0$: $y = c$

x-intercepts (zeros/roots):

• Set $y = 0$ and solve quadratic

Vertex Form: $y = a(x - h)^2 + k$

Vertex: $(h, k)$ — you can read it directly!

Example: $y = 2(x - 3)^2 + 1$

• Vertex: $(3, 1)$
• Opens up ($a = 2 > 0$)
• Narrower than $y = x^2$ ($|a| = 2 > 1$)

SAT Question Types

Type 1: Solve the Quadratic

Strategy:

• Try factoring first (fastest)
• Use quadratic formula if doesn't factor
• Check answer choices (can plug them in!)

Type 2: Number of Solutions

Strategy:

• Calculate discriminant $b^2 - 4ac$
• Positive = 2, Zero = 1, Negative = 0

Type 3: Find the Vertex

Strategy:

• If in vertex form $a(x-h)^2 + k$: vertex is $(h, k)$
• If in standard form: $x = -\frac{b}{2a}$, then find $y$

Type 4: Graph Interpretation

Strategy:

• Check where graph crosses x-axis (solutions)
• Find highest/lowest point (vertex)
• Determine direction (up or down)

Type 5: Word Problems

Common scenarios:

• Projectile motion: $h(t) = -16t^2 + v_0 t + h_0$
• Area problems: $A = x(20 - x)$

Strategy:

• Set up equation from problem
• Solve using appropriate method
• Check answer makes sense in context

Factoring Review

Common Patterns

Difference of Squares: $x^2 - a^2 = (x + a)(x - a)$

Example: $x^2 - 25 = (x + 5)(x - 5)$

Perfect Square Trinomials: $x^2 + 2ax + a^2 = (x + a)^2$ $x^2 - 2ax + a^2 = (x - a)^2$

Example: $x^2 + 10x + 25 = (x + 5)^2$

Standard Factoring: $x^2 + bx + c = (x + m)(x + n)$

where $m \cdot n = c$ and $m + n = b$

Example: $x^2 + 7x + 12$

• Need: product = 12, sum = 7
• Numbers: 3 and 4
• Answer: $(x + 3)(x + 4)$

Common SAT Mistakes

❌ Forgetting ± in square root method
$x^2 = 9$ → $x = \pm 3$, NOT just $x = 3$

❌ Arithmetic errors in quadratic formula
Be careful with negatives and parentheses!

❌ Not setting equation to zero before factoring
Must have 0 on one side

❌ Confusing vertex form $(x-h)^2$ with $(x+h)^2$
$y = (x - 3)^2$ has vertex at $x = 3$, not $x = -3$

❌ Forgetting to find BOTH coordinates of vertex
Need both $x$ and $y$ values

Quick Tips

✓ Try factoring first — it's fastest when it works
✓ Quadratic formula always works — reliable backup
✓ Check by plugging answer back in — catches mistakes
✓ Graph on calculator if allowed — visual confirmation
✓ Remember ± when taking square roots
✓ Discriminant saves time for "how many solutions" questions

Practice Approach

1. Identify what the question asks (solve, vertex, number of solutions, etc.)
2. Choose method based on what's given and what's asked
3. Execute carefully (factoring, formula, or graph)
4. Check answer makes sense (plug back in, check with graph)
5. Verify units/context for word problems

Remember: Quadratics always have at most 2 solutions, and the SAT loves testing whether you remember the ± sign!