Factor 4x^2+2x-3

Problem

4*x2+2*x−3

Solution

1. Identify the coefficients of the quadratic expression in the form a*x2+b*x+c Here, a=4 b=2 and c=−3

2. Calculate the product a*c which is 4×−3=−12

3. Seek two numbers that multiply to −12 and add to the middle coefficient b=2

4. List the factors of −12 (−1,12) (1,−12) (−2,6) (2,−6) (−3,4) (3,−4)

5. Check the sums of these pairs: −1+12=11 1−12=−11 −2+6=4 2−6=−4 −3+4=1 3−4=−1

6. Conclude that there are no integers that satisfy both conditions, as no pair of factors of −12 sums to 2

7. Apply the quadratic formula to find the roots of the related equation 4*x2+2*x−3=0 to determine if it can be factored over the real numbers.

8. Calculate the roots using x=(−b±√(,b2−4*a*c))/(2*a)

x=(−2±√(,2−4*(4)*(−3)))/(2*(4))

x=(−2±√(,4+48))/8

x=(−2±√(,52))/8

x=(−2±2√(,13))/8

x=(−1±√(,13))/4

9. Determine that since the roots are irrational, the expression 4*x2+2*x−3 is irreducible over the integers.

Final Answer

4*x2+2*x−3=4*x2+2*x−3

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