First try to solve the equation by factoring. If you are unable to solve the equation by factoring, solve the equation by using the quadratic formula. (Enter your answers as a comma-separated list. If the solution is not a real number, enter NOT REAL.)
To solve a quadratic equation, you can first attempt to factor it. Let's say we have a quadratic equation in the form of ax^2 + bx + c = 0.
1. Factorization Method:
First, try to factor the equation into two binomial expressions. This involves finding two numbers that multiply to give the constant term (c) and add up to give the coefficient of the middle-term (b). For instance, let's consider the equation: 2x^2 + 5x + 3 = 0.
To factor this equation, we need to find two numbers that multiply to give 3 (constant term) and add up to give 5 (coefficient of x). In this case, the numbers are 3 and 1, since 3 * 1 = 3 and 3 + 1 = 4. So, we can rewrite the equation as: 2x^2 + 3x + 2x + 3 = 0.
Now, we can group the terms as follows: (2x^2 + 3x) + (2x + 3) = 0.
Taking out the common factors, we get: x(2x + 3) + 1(2x + 3) = 0.
Notice that we have a common binomial factor, (2x + 3), which allows us to rewrite the equation as: (2x + 3)(x + 1) = 0.
Now, we can set each factor equal to zero and solve for x:
2x + 3 = 0 => 2x = -3 => x = -3/2.
x + 1 = 0 => x = -1.
So, the solutions to the equation 2x^2 + 5x + 3 = 0 are x = -3/2 and x = -1.
If factoring is not possible, you can move on to the Quadratic Formula.
2. Quadratic Formula:
The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation.
For example, let's consider the quadratic equation: 3x^2 - 7x + 2 = 0.
By comparing the equation with the standard form ax^2 + bx + c = 0, we can determine that a = 3, b = -7, and c = 2.
Using the quadratic formula, we can plug in these values:
x = (-(-7) ± √((-7)^2 - 4(3)(2))) / (2(3))
= (7 ± √(49 - 24)) / 6
= (7 ± √25) / 6.
Now, we have two possibilities:
For the positive square root, x = (7 + 5) / 6 = 12 / 6 = 2.
For the negative square root, x = (7 - 5) / 6 =2 / 6 = 1/3.
Thus, the solutions to the equation 3x^2 - 7x + 2 = 0 are x = 2 and x = 1/3.
Remember, if the discriminant (b^2 - 4ac) is negative, the solutions to the quadratic equation will be complex or imaginary, denoted as NOT REAL.
Hope this explanation guides you through solving quadratic equations using factoring and the quadratic formula!