Solve: (2x – 3)(3x – 5) = 0
Solve by the quadratic formula. Leave the square root in your answer:
3x2 – x – 8 = 0
Factor the first one by setting each term equal to 0.
2x - 3 = 0
2x = 3
x = 3/2
Do the same process for 3x - 5.
For the second, look over the Quadratic Formula. It states that for an equation ax^2 + bx + c, x will equal the opposite of b plus or minus the square root of b squared minus 4ac all over 2a. (Look it up for the formula equation. It's difficult to type.) All you have to do is identify a, b, and c, and then plug it into the QF. You will have two answers, one using the plus and one using the minus.
To solve the quadratic equation (2x – 3)(3x – 5) = 0, we can use the zero-product property. According to the zero-product property, if the product of two factors is zero, then at least one of those factors must be zero.
This means that either (2x – 3) = 0 or (3x – 5) = 0. We can solve each equation separately to find the values of x.
For (2x – 3) = 0, add 3 to both sides and then divide both sides by 2:
2x – 3 + 3 = 0 + 3
2x = 3
x = 3/2
For (3x – 5) = 0, add 5 to both sides and then divide both sides by 3:
3x – 5 + 5 = 0 + 5
3x = 5
x = 5/3
So the two solutions to the equation (2x – 3)(3x – 5) = 0 are x = 3/2 and x = 5/3.
Now, let's move on to the second equation: 3x^2 – x – 8 = 0. To solve this equation using the quadratic formula, we need to determine the coefficients for the quadratic equation: a, b, and c.
In this case, a is the coefficient of x^2, which is 3.
b is the coefficient of x, which is -1.
c is the constant term, which is -8.
According to the quadratic formula, the solutions for a quadratic equation ax^2 + bx + c = 0 are given by the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values for a, b, and c from the equation 3x^2 – x – 8 = 0, we have:
x = (-(-1) ± √((-1)^2 - 4(3)(-8))) / (2(3))
x = (1 ± √(1 + 96)) / 6
x = (1 ± √97) / 6
So the solutions to the equation 3x^2 – x – 8 = 0 are x = (1 + √97) / 6 and x = (1 - √97) / 6.