Solve Quadratics by
Multiple Methods
Master five powerful techniques to solve any quadratic equation: inspection, square roots, factoring, completing the square, and the quadratic formula.
Algebra 2
What Is a Quadratic Equation?
A quadratic equation is any equation that can be written in standard form:
The variable \( x \) appears with a highest power of 2. The constants \( a \), \( b \), and \( c \) are real numbers. The solutions (also called roots or zeros) are the \( x \)-values that make the equation true.
Visualizing Roots on a Parabola
The Five Methods at a Glance
1. Inspection
Recognize the answer by sight: \( x^2 = 49 \)
2. Square Roots
Isolate \( x^2 \), then take \( \pm\sqrt{\phantom{x}} \)
3. Factoring
Write as a product of two binomials, set each to 0
4. Completing the Square
Rewrite as \( (x - h)^2 = k \), then take roots
5. Quadratic Formula
Works for every quadratic: always gives the answer
The Quadratic Formula
For any quadratic \( ax^2 + bx + c = 0 \) with \( a \neq 0 \):
The expression under the radical, \( \Delta = b^2 - 4ac \), is called the discriminant. It tells you how many real solutions exist.
\( \Delta > 0 \)
Two distinct real roots
\( \Delta = 0 \)
One repeated real root
\( \Delta < 0 \)
Two complex roots: \( a \pm bi \)
The Discriminant Decides Everything
Factoring vs. Quadratic Formula
| Factoring | Quadratic Formula | |
|---|---|---|
| Speed | Very fast when it works | Moderate: always reliable |
| When to use | Integer or simple rational roots | Any quadratic: always works |
| Handles complex roots? | No | Yes |
| Error risk | Sign errors in factors | Arithmetic under the radical |
| Best for | \( x^2 + 5x + 6 = 0 \) | \( 3x^2 - 2x + 5 = 0 \) |
How to Choose the Right Method
1
Is \( b = 0 \)? Use square roots: isolate \( x^2 \), then take \( \pm\sqrt{\phantom{x}} \).
2
Can you spot the answer? Use inspection: e.g., \( x^2 = 49 \Rightarrow x = \pm 7 \).
3
Does it factor easily? Look for two numbers that multiply to \( ac \) and add to \( b \).
4
Is vertex form useful? Use completing the square when the problem asks for vertex form or exact radicals.
5
None of the above? The quadratic formula always works: plug in \( a \), \( b \), \( c \) and simplify.
Example 1: Factoring
Solve \( x^2 + 5x + 6 = 0 \) by factoring.
1
Find two numbers that multiply to 6 and add to 5: the numbers are 2 and 3.
2
Write the factored form: \( (x + 2)(x + 3) = 0 \).
3
Set each factor to zero: \( x + 2 = 0 \) or \( x + 3 = 0 \).
4
Solve: \( x = -2 \) or \( x = -3 \).
Example 2: Square Roots
Solve \( 3x^2 - 75 = 0 \).
1
Isolate \( x^2 \): add 75 to both sides, then divide by 3: $$ x^2 = 25 $$
2
Take the square root of both sides. Remember the \( \pm \): $$ x = \pm\sqrt{25} = \pm 5 $$
Example 3: Completing the Square
Solve \( x^2 + 6x + 2 = 0 \) by completing the square.
1
Move the constant: \( x^2 + 6x = -2 \).
2
Take half of 6, square it: \( \left(\frac{6}{2}\right)^2 = 9 \). Add 9 to both sides: $$ x^2 + 6x + 9 = 7 $$
3
Factor the left side as a perfect square: \( (x + 3)^2 = 7 \).
4
Take square roots: \( x + 3 = \pm\sqrt{7} \), so \( x = -3 \pm \sqrt{7} \).
Example 4: The Quadratic Formula
Solve \( 2x^2 - 4x - 3 = 0 \) using the quadratic formula.
1
Identify: \( a = 2 \), \( b = -4 \), \( c = -3 \).
2
Compute the discriminant: $$ \Delta = (-4)^2 - 4(2)(-3) = 16 + 24 = 40 $$
3
Substitute into the formula: $$ x = \frac{-(-4) \pm \sqrt{40}}{2(2)} = \frac{4 \pm 2\sqrt{10}}{4} = \frac{2 \pm \sqrt{10}}{2} $$
Example 5: Complex Roots
Solve \( x^2 + 2x + 5 = 0 \).
1
Identify: \( a = 1 \), \( b = 2 \), \( c = 5 \).
2
Discriminant: \( \Delta = b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16 \). Since \( \Delta < 0 \), the roots are complex.
3
Apply the formula: $$ x = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i $$
Common Mistakes to Avoid
✕ Forgetting \( \pm \)
When taking square roots, always write both the positive and negative root: \( x = \pm\sqrt{k} \).
✕ Sign errors in the formula
The formula has \( -b \), not \( b \). When \( b \) is negative, \( -b \) is positive. Use parentheses carefully.
✕ Wrong discriminant
Remember: \( b^2 - 4ac \), not \( b^2 - 4a \) or \( b^2 + 4ac \). Include the minus and multiply all three: \( 4 \cdot a \cdot c \).
✕ Not simplifying radicals
Always simplify: \( \sqrt{40} = 2\sqrt{10} \). Then reduce the entire fraction by dividing numerator and denominator.
Method Summary
| Method | Best When | Key Step |
|---|---|---|
| Inspection | \( x^2 = k \) (perfect square) | Recognize \( x = \pm\sqrt{k} \) |
| Square Roots | \( b = 0 \): no linear term | Isolate \( x^2 \), take \( \pm\sqrt{\phantom{x}} \) |
| Factoring | Integer/simple roots | Find factors of \( ac \) that add to \( b \) |
| Completing Square | \(a=1\) and \(b\) is even | Add \( \left(\frac{b}{2}\right)^2 \) to both sides |
| Quadratic Formula | Always works | Plug into \( x = \frac{-b\pm\sqrt{\Delta}}{2a} \) |
Question 1 of 5
Solve \( x^2 - 9 = 0 \). What are the solutions?
Since \( x^2 = 9 \), taking the square root of both sides gives \( x = \pm\sqrt{9} = \pm 3 \). Always include both the positive and negative root.
Question 2 of 5
What is the discriminant of \( 2x^2 + 3x - 5 = 0 \)?
The discriminant is \( b^2 - 4ac = 3^2 - 4(2)(-5) = 9 + 40 = 49 \). Since \( 49 > 0 \), there are two distinct real roots.
Question 3 of 5
Solve by factoring: \( x^2 - x - 12 = 0 \).
We need two numbers that multiply to \( -12 \) and add to \( -1 \): those are \( -4 \) and \( 3 \). So \( (x - 4)(x + 3) = 0 \), giving \( x = 4 \) or \( x = -3 \).
Question 4 of 5
Solve \( x^2 + 4x + 13 = 0 \). What are the solutions?
Here \( a=1, b=4, c=13 \). The discriminant is \( \Delta = b^2 - 4ac = 4^2 - 4(1)(13) = 16 - 52 = -36 \). So \( x = \frac{-4 \pm \sqrt{-36}}{2} = \frac{-4 \pm 6i}{2} = -2 \pm 3i \). Complex roots are written in the form \( a \pm bi \).
Question 5 of 5
Which method is most efficient to solve \( 5x^2 - 80 = 0 \)?
Since there is no linear term (\( b = 0 \)), the fastest method is to isolate \( x^2 \): divide by 5 to get \( x^2 = 16 \), then \( x = \pm 4 \). No factoring or formula needed!