Solve the quadratic equation 2x² - 4x + 1 = 0
How many solutions are in the equation 3x² + 4x - 7 = 0
(a) use the quadratic formula
(b) the discriminant is positive, so two real roots
1. 2x^2 - 4x + 1 = 0,
X = (-B +- sqrt(B^2-4AC))/2A,
X = (4 +- sqrt(16 - 8))/4 = 1.71, and 0.294.
2. 3x^2 + 4x - 7 = 0,
B^2 - 4AC = 16 + 84 = 100,
The discriminant is positive; therefore, we have 2 real solutions.
To solve the quadratic equation 2x² - 4x + 1 = 0, we can use the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
Here, a = 2, b = -4, and c = 1.
Plugging these values into the formula, we get:
x = (-(-4) ± √((-4)² - 4(2)(1))) / (2(2))
x = (4 ± √(16 - 8)) / 4
x = (4 ± √8) / 4
x = (4 ± 2√2) / 4
Simplifying further, we get two solutions:
x₁ = (4 + 2√2) / 4
x₂ = (4 - 2√2) / 4
Reducing the fractions, we get:
x₁ = 1 + √2/2
x₂ = 1 - √2/2
Therefore, the solutions to the quadratic equation 2x² - 4x + 1 = 0 are x = 1 + √2/2 and x = 1 - √2/2.
Now, let's solve the equation 3x² + 4x - 7 = 0 to determine how many solutions it has.
Using the quadratic formula again, we identify:
a = 3, b = 4, and c = -7.
Plugging in these values, we obtain:
x = (-4 ± √(4² - 4(3)(-7))) / (2(3))
x = (-4 ± √(16 + 84)) / 6
x = (-4 ± √100) / 6
x = (-4 ± 10) / 6
Simplifying further, we get two solutions:
x₁ = (-4 + 10) / 6
x₂ = (-4 - 10) / 6
Reducing the fractions, we arrive at:
x₁ = 6/6
x₂ = -14/6
Simplifying these fractions, we have:
x₁ = 1
x₂ = -7/3
Therefore, the equation 3x² + 4x - 7 = 0 has two solutions: x = 1 and x = -7/3.
To solve the quadratic equation 2x² - 4x + 1 = 0, we can use the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
Comparing the equation to the standard quadratic form ax² + bx + c = 0, we can see that a = 2, b = -4, and c = 1. Plugging these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)² - 4(2)(1))) / (2(2))
= (4 ± √(16 - 8)) / 4
= (4 ± √8) / 4
Simplifying further, we can write the solutions as:
x = (4 + √8) / 4
x = (4 - √8) / 4
Now, let's solve the equation 3x² + 4x - 7 = 0 to determine how many solutions it has.
Applying the quadratic formula again, we have:
x = (-b ± √(b² - 4ac)) / (2a)
Comparing the equation to the standard form, we can see that a = 3, b = 4, and c = -7. Plugging these values into the quadratic formula, we get:
x = (-(4) ± √((4)² - 4(3)(-7))) / (2(3))
= (-4 ± √(16 + 84)) / 6
= (-4 ± √100) / 6
= (-4 ± 10) / 6
Simplifying further, we get the solutions as:
x = (-4 + 10) / 6
x = (6) / 6
x = 1
and
x = (-4 - 10) / 6
x = (-14) / 6
x = -7/3
Thus, the given equation 3x² + 4x - 7 = 0 has two solutions, x = 1 and x = -7/3.