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Part 1
Solve the equation for real solutions by the quadratic formula.
3x2−5x−7=0
To solve the equation 3x^2 - 5x - 7 = 0 using the quadratic formula, we first identify the coefficients a, b, and c in the general quadratic equation format ax^2 + bx + c = 0.
Comparing the given equation 3x^2 - 5x - 7 = 0 to the general format, we have:
a = 3
b = -5
c = -7
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c, we can solve for x:
x = (-(-5) ± √((-5)^2 - 4(3)(-7))) / (2(3))
Simplifying further:
x = (5 ± √(25 + 84)) / 6
x = (5 ± √(109)) / 6
Therefore, the solutions for the equation 3x^2 - 5x - 7 = 0 are:
x = (5 + √109) / 6
x = (5 - √109) / 6
To solve the quadratic equation 3x^2 - 5x - 7 = 0 using the quadratic formula, follow these steps:
Step 1: Identify the coefficients.
The quadratic equation is in the standard form: ax^2 + bx + c = 0.
In this equation, a = 3, b = -5, and c = -7.
Step 2: Write down the quadratic formula.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Step 3: Substitute the coefficients into the formula.
Substitute the values of a, b, and c into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4(3)(-7))) / (2(3))
Step 4: Simplify inside the square root.
Simplify the square root part of the formula:
x = (5 ± √(25 + 84)) / 6
= (5 ± √109) / 6
Step 5: Evaluate both solutions.
Evaluate both solutions by considering the plus and minus signs separately:
x1 = (5 + √109) / 6
and
x2 = (5 - √109) / 6
Therefore, the real solutions to the equation 3x^2 - 5x - 7 = 0 are x1 = (5 + √109) / 6 and x2 = (5 - √109) / 6.
To solve the quadratic equation 3x^2 - 5x - 7 = 0 using the quadratic formula, you will need to follow these steps:
Step 1: Identify the coefficients of the variables in the equation. In this case, the coefficient of x^2 is 3, the coefficient of x is -5, and the constant term is -7.
Step 2: Plug the values into the quadratic formula, which is:
x = (-b ± √(b^2 - 4ac)) / (2a)
Step 3: Substitute the values from the equation into the quadratic formula. In this case, a = 3, b = -5, and c = -7:
x = (-(-5) ± √((-5)^2 - 4(3)(-7))) / (2(3))
Step 4: Simplify the expression inside the square root:
x = (-(-5) ± √(25 + 84)) / (2(3))
x = (5 ± √109) / 6
Therefore, the two real solutions for the equation 3x^2 - 5x - 7 = 0 are:
x = (5 + √109) / 6
x = (5 - √109) / 6