Explain why we know that 3x^2+4x−2 is not factorable. but make it look like a 10th grader did it
We know that 3x^2+4x−2 is not factorable because it does not have two binomial factors that can be multiplied together to give us this equation. If it were factorable, we could write it as (ax+b)(cx+d) and get back to the original equation when we multiply it out. But when we try to find these binomial factors for 3x^2+4x−2, we see that it's not possible because there is no combination of numbers that can give us the coefficients of x^2, x, and the constant term −2. Therefore, we can conclude that 3x^2+4x−2 is not factorable.
To determine if the quadratic expression 3x^2+4x−2 is factorable, we can use the discriminant. The formula to find the discriminant is b^2−4ac. We have a=3, b=4, and c=-2.
Step 1: Substitute these values into the discriminant formula:
Discriminant = (4)^2−4(3)(-2)
Step 2: Simplify the expression inside the parentheses:
Discriminant = 16−4(3)(-2)
Step 3: Simplify further by following the order of operations, which is parentheses first, then multiplication:
Discriminant = 16+24
Step 4: Add the numbers together:
Discriminant = 40
Step 5: Check the value of the discriminant. Since the discriminant is positive (40>0), this tells us that the quadratic expression 3x^2+4x−2 has two distinct real roots.
Therefore, since the discriminant is positive, the quadratic expression 3x^2+4x−2 is not factorable.
To determine if the equation 3x^2+4x−2 is factorable, we can use the quadratic formula. The quadratic formula is x = (-b ± √(b^2-4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation (in this case, a = 3, b = 4, and c = -2). This formula allows us to find the roots or solutions of the equation.
Substituting the values into the quadratic formula, we have x = (-4 ± √(4^2-4(3)(-2))) / (2(3)). Simplifying this further, we get x = (-4 ± √(16+24)) / 6. Further simplification gives x = (-4 ± √40) / 6.
To determine if the equation is factorable, we need to check if the term inside the square root can be simplified as a perfect square. In this case, the square root of 40 cannot be written as an integer. Hence, we cannot simplify the term and the quadratic equation 3x^2+4x−2 is not factorable with integer coefficients.