Solve the quadratic equation using factorization: x2 - 5x + 6 = 0.
Get the complete, step-by-step math solution for: "Solve the quadratic equation using factorization: x² - 5x + 6 = 0.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify coefficients and find factors
We need to factor the quadratic equation x2−5x+6=0. This is in the standard form ax2+bx+c=0, where a=1, b=−5, and c=6. We need to find two numbers that multiply to c (which is 6) and add up to b (which is -5).
Step 2: Find the two numbers
The two numbers that satisfy these conditions are -2 and -3, because their product is (−2)×(−3)=6 and their sum is (−2)+(−3)=−5.
Step 3: Rewrite the middle term
Now, we rewrite the middle term, −5x, using these two numbers: -2x - 3x. This allows us to group terms for factorization.
Step 4: Factor by grouping
We group the terms and factor out the common factors from each pair. From the first two terms, x2−2x, we factor out x. From the last two terms, −3x+6, we factor out −3.
Step 5: Factor out the common binomial
Notice that (x−2) is a common binomial factor in both terms. We factor it out to get the completely factored form of the quadratic equation.
Step 6: Solve for x
For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for x.