Solve -2x^2+4x+1
Solve ax² + bx + c = 0 using the quadratic formula with discriminant analysis.
-2x2+4x+1
Solve using the quadratic formula since this quadratic does not factor nicely:
Multiply by −1:
2x2 − 4x − 1 = 0
Compare with the standard form ax² + bx + c = 0:
a = 2, b = −4, c = −1
x = (−b ± √b2 − 4ac) / 2a
x = (4 ± √(-4)2 − 4(2)(-1)) / 2(2)
x = (4 ± √16 + 8) / 4
x = (4 ± √24) / 4
Δ = b² − 4ac = 16 − -8 = 24
Since Δ = 24 is positive but not a perfect square, the roots are irrational.
Simplify √24 = 2√6:
x = (4 ± 2√6) / 4
x = (2 ± √6) / 2
x1 = (2 + √6) / 2 ≈ 2.2247
x2 = (2 − √6) / 2 ≈ -0.2247
x = 1 - √62 ≈ -0.2247, 1 + √62 ≈ 2.2247
How to solve -2x^2+4x+1
To solve -2x^2+4x+1, we can try factoring, use the quadratic formula, or complete the square. The solution is x = 1 - √(6)/2 ≈ -0.2247, 1 + √(6)/2 ≈ 2.2247.
This is a second-degree quadratic equation — the highest power of the variable is 2. Quadratic equations can have two real roots, one repeated root, or two complex roots.
Frequently asked questions
What is the answer to -2x^2+4x+1?
The answer is x = 1 - √(6)/2 ≈ -0.2247, 1 + √(6)/2 ≈ 2.2247.
What method is used?
factoring when possible, or the quadratic formula x = (−b ± √(b²−4ac)) / 2a. The discriminant b²−4ac determines the number and type of solutions.
How do I verify this?
Substituting the solution(s) back into -2x^2+4x+1 confirms both sides are equal.