Complete the Square for 4x^2+11x-3
Complete the square for the quadratic:
4x2+11x-3
Set up the a, b, and c values:
a = 4, b = 11, c = 1
Subtract 1 to each side
4x2 + 11x + 1 - 1 = 0 - 1
x2x = -1
Since our a coefficient of 4 ≠ 1
We divide our equation by 4
x2 + 11/4 = -1/4
Complete the square:
Add an amount to both sides
x2 + 11/4x + ? = -1/4 + ?
Add (½*middle coefficient)2 to each side
| Amount to add = | (1 x 11)2 |
| (2 x 4)2 |
| Amount to add = | (11)2 |
| (8)2 |
| Amount to add = | 121 |
| 64 |
Amount to add = 121/64
Rewrite our perfect square equation:
x2 + 11/4 + (11/8)2 = -1/4 + (11/8)2
(x + 11/8)2 = -1/4 + 121/64
Simplify Right Side of the Equation:
We multiply -1 by 64 ÷ 4 = 16 and 121 by 64 ÷ 64 = 1
| Simplified Fraction = | -1 x 16 + 121 x 1 |
| 64 |
| Simplified Fraction = | -16 + 121 |
| 64 |
| Simplified Fraction = | 105 |
| 64 |
Our fraction can be reduced down:
Using our GCF of 105 and 64 = 105
Reducing top and bottom by 105 we get
1/0.60952380952381
We set our left side = u
u2 = (x + 11/8)2
u has two solutions:
u = +√1/0.60952380952381
u = -√1/0.60952380952381
Replacing u, we get:
x + 11/8 = +1
x + 11/8 = -1
Subtract 11/8 from the both sides
x + 11/8 - 11/8 = +1/1 - 11/8
Simplify right side of the equation
We multiply 1 by 8 ÷ 1 = 8 and -11 by 8 ÷ 8 = 1
| Simplified Fraction = | 1 x 8 - 11 x 1 |
| 8 |
| Simplified Fraction = | 8 - 11 |
| 8 |
| Simplified Fraction = | -3 |
| 8 |
Answer 1 = -3/8
Subtract 11/8 from the both sides
x + 11/8 - 11/8 = -1/1 - 11/8
Simplify right side of the equation
We multiply -1 by 8 ÷ 1 = 8 and -11 by 8 ÷ 8 = 1
| Simplified Fraction = | -1 x 8 - 11 x 1 |
| 8 |
| Simplified Fraction = | -8 - 11 |
| 8 |
| Simplified Fraction = | -19 |
| 8 |
Answer 2 = -19/8
Final Answer
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.
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
What 9 concepts are covered in the Quadratic Equations and Inequalities Calculator?
complete the squarea technique for converting a quadratic polynomial of the form ax2 + bx + c to a(x - h)2 + kequationa statement declaring two mathematical expressions are equalfactora 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.interceptparabolaa plane curve which is approximately U-shapedquadraticPolynomials with a maximum term degree as the second degreequadratic equations and inequalitiesrational rootvertexHighest point or where 2 curves meetExample calculations for the Quadratic Equations and Inequalities Calculator
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