Use factoring to solve the quadratic equation.
5x2 - 25x + 30 = 0
Divide both sides by 5 first. It makes it easier.
x^2 -5x +6 = 0
(x-3)(x-2) = 0
x = 2 or 3
To solve the quadratic equation 5x^2 - 25x + 30 = 0 using factoring, we need to factor the expression on the left side of the equation and set each factor equal to zero.
Step 1: Start by looking for two numbers that multiply to give the product of the coefficient of x^2 (5) and the constant term (30), and also add up to give the coefficient of x (-25).
The factors of 5 are 1 and 5, and the factors of 30 are 1, 2, 3, 5, 6, 10, 15, and 30.
By experimenting, we find that the numbers -5 and -6 fit the criteria. They multiply to give 30 and add up to -25: -5 * -6 = 30 and -5 + (-6) = -25.
Step 2: Rewrite the middle term -25x as the sum of -5x and -20x. We split the -25x term using the numbers found in step 1.
So, the equation becomes: 5x^2 - 5x - 20x + 30 = 0.
Step 3: Group the terms in pairs and factor by grouping.
Group the first two terms 5x^2 and -5x: 5x(x - 1)
Group the last two terms -20x and 30: -10(x - 3)
So, the equation becomes: 5x(x - 1) - 10(x - 3) = 0.
Step 4: Factor out the common factors from each group.
In the first group, the common factor is x, so we have: x(5x - 5).
In the second group, the common factor is -10, so we have: -10(x - 3).
Now, the equation becomes: x(5x - 5) - 10(x - 3) = 0.
Step 5: Simplify the equation by distributing.
Distribute the x to each term inside the parentheses and the -10 to each term inside the other parentheses: 5x^2 - 5x - 10x + 30 = 0.
Combine like terms: 5x^2 - 15x + 30 = 0.
Step 6: Factor out any common factors, if possible.
In this case, there are no common factors, so we can continue to solve the equation.
Step 7: Set each factor equal to zero and solve for x.
5x^2 - 15x + 30 = 0 can be rewritten as 5(x^2 - 3x + 6) = 0.
Setting each factor equal to zero, we have:
x^2 - 3x + 6 = 0.
Now, we can solve for x by either factoring or using the quadratic formula.
Unfortunately, in this case, the quadratic equation x^2 - 3x + 6 cannot be factored further, so we need to use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a.
For our quadratic equation x^2 - 3x + 6 = 0, the coefficients are a = 1, b = -3, and c = 6.
Calculating the discriminant (b^2 - 4ac):
D = (-3)^2 - 4(1)(6) = 9 - 24 = -15.
Since the discriminant is negative (-15 < 0), the quadratic equation has no real solutions.