Solve the quadratic:
w2+3w-108 = 0
Set up the a, b, and c values:
a = 1, b = 3, c = -108
Quadratic Formula
| w = | -b ± √b2 - 4ac |
| 2a |
Calculate -b
-b = -(3)
-b = -3
Calculate the discriminant Δ
Δ = b2 - 4ac:
Δ = 32 - 4 x 1 x -108
Δ = 9 - -432
Δ = 441 <--- Discriminant
Since Δ > 0, we expect two real roots.
Take the square root of Δ
√Δ = √(441)
√Δ = 21
-b + Δ:
Numerator 1 = -b + √Δ
Numerator 1 = -3 + 21
Numerator 1 = 18
-b - Δ:
Numerator 2 = -b - √Δ
Numerator 2 = -3 - 21
Numerator 2 = -24
Calculate 2a
Denominator = 2 * a
Denominator = 2 * 1
Denominator = 2
Find Solutions
| Solution 1 = | Numerator 1 |
| Denominator |
| Solution 1 = | 18 |
| 2 |
Solution 1 = 9
Solution 2
| Solution 2 = | Numerator 2 |
| Denominator |
| Solution 2 = | -24 |
| 2 |
Solution 2 = -12
Solution Set
(Solution 1, Solution 2) = (9, -12)
Prove our first answer
(9)2 + 3(9) - 108 ? 0
(81) + 27108 ? 0
81 + 27108 ? 0
0 = 0
Prove our second answer
(-12)2 + 3(-12) - 108 ? 0
(144) - 36108 ? 0
144 - 36108 ? 0
0 = 0
Final Answer
(Solution 1, Solution 2) = (9, -12)
Keep Practicing
Common Core State Standards In This Lesson
HSN.CN.C.7, HSA.SSE.B.3.A, HSA.SSE.B.3.B, HSA.REI.B.4, HSA.REI.B.4.A, HSF.IF.C.8.A
What is the Answer?
(Solution 1, Solution 2) = (9, -12)
How does the Quadratic Equations and Inequalities Calculator work?
Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax2 + bx + c = 0. Also generates practice problems as well as hints for each problem.
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for the Quadratic
* Factor the Quadratic
* Y-Intercept
* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)2 + k
* Concavity of the parabola formed by the quadratic
* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.
This calculator has 4 inputs.
* Solve using the quadratic formula and the discriminant Δ
* Complete the Square for the Quadratic
* Factor the Quadratic
* Y-Intercept
* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)2 + k
* Concavity of the parabola formed by the quadratic
* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.
This calculator has 4 inputs.
What 5 formulas are used for the Quadratic Equations and Inequalities Calculator?
y = ax2 + bx + c
(-b ± √b2 - 4ac)/2a
h (Axis of Symmetry) = -b/2a
The vertex of a parabola is (h,k) where y = a(x - h)2 + k
For more math formulas, check out our Formula Dossier
(-b ± √b2 - 4ac)/2a
h (Axis of Symmetry) = -b/2a
The vertex of a parabola is (h,k) where y = a(x - h)2 + k
For more math formulas, check out our Formula Dossier
What 9 concepts are covered in the Quadratic Equations and Inequalities Calculator?
- complete the square
- a technique for converting a quadratic polynomial of the form ax2 + bx + c to a(x - h)2 + k
- equation
- a statement declaring two mathematical expressions are equal
- factor
- a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n.
- intercept
- parabola
- a plane curve which is approximately U-shaped
- quadratic
- Polynomials with a maximum term degree as the second degree
- quadratic equations and inequalities
- rational root
- vertex
- Highest point or where 2 curves meet
Example calculations for the Quadratic Equations and Inequalities Calculator
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