Answer:
Master product = 36
I'll also add the solutions to the quadratic equation just in case you might need it later:
x = 1 and x = 9/4 (aka 2.25)
Explanation:
Currently, 4x^2 - 13x + 9 = 0 is in standard form, whose general equation is
ax^2 + bx + c. The master product is a set in solving by factoring, where we look for two numbers whose product equals a * c and add up to b.
The a * c is what we call the master product.
Thus, since 4 is a and 9 is c, the master product of our equation is 36 since 4 * 9 = 36
Do you have to solve for x or do you just need the master product. I'll find x anyhow, but if you just need the master product, just write that.
Steps for solving:
Step 1: Make sure trinomial is in standard form:
4x^2 - 13x + 9 = 0 is already in standard form, so this step is already given/complete.
Step 2: Find two terms whose product equals a*c and adds up to b:
In our quadratic, 4 is a, -13 is b, and 9 is c. We see that 4 * 9 = 36 and -4 * -9 = 36. Also, -4 + (-9) = -13, so our two numbers are -4 and -9.
Step 3: Replace bx term with two bx terms that use the numbers from:
Since our two numbers from step 2 are -4 and -9, we can replace -13x with -4x and -9x. Thus, our new quadratic is 4x^2 - 4x - 9x + 9
Step 4: Factor by grouping:
We can group 4x^2 -4x and -9x + 9
The greatest common factor of 4x^2 and -4x is 4x. Factoring it out gives us 4x(x - 1).
The greatest common factor of -9x and 9 is -9. Factoring it out gives us
-9(x - 1).
Step 5: Factor out common term and remaining term.
x - 1 is the common term, which leaves us with 4x - 9.
Thus, our equation is now (x - 1)(4x - 9) = 0.
Step 6: Set each term equal o 0 to solve for x:
Setting (x-1) equal to 0 and solving:
x - 1 = 0
x = 1
Setting 4x - 9 equal to 0 and solving:
4x - 9 = 0
4x = 9
x = 9/4
Thus, the solutions to the quadratic equation are x = 1 and x = 9/4.
Optional Step 7: Check validity of solutions by plugging in 1 for x and 9/4 for x in original equation in standard form:
Plugging in 1 for x in 4x^2 - 13x + 9 = 0:
4(1)^2 - 13(1) + 9 = 0
4(1) - 13 + 9 = 0
4 - 13 + 9 = 0
-9 + 9 = 0
0 = 0
Plugging in 9/4 for x in 4x^2 - 13 + 9 = 0:
4(9/4)^x - 13(9/4) + 9 = 0
4(81/16) - 117/4 + 9 = 0
81/4 - 117/4 + 9 = 0
-9 + 9 = 0
0 = 0
Thus, our solutions are correct