I need help on these problems. There over solving equations by factoring.
1. 4x^2=9
2. M^2-36=16M
3. R^2+9=10r
4. 6s^2=17s-12
set everything to zero, then you can factor.
#4
6s^2-17s+12 = 0
You know the factors will look like
(6x-?)(x-?) or (3x-?)(2x-?)
So, since -17 is fairly large, we will probably have the first
(6s-3)(s-4) gives -27s too big
(6s-4)(s-3) gives -22s still too big
(6s-6)(s-2) gives -18s close, but no cigar
(3s-4)(2s-3) gives -17s YES!
Do the others the same way. #1 involves a difference of squares, so it's easy.
Would #4 be s=1 and s=2?
well, no.
If the product of some numbers is zero, then one of the numbers must be zero. We have
(3s-4)(2s-3) = 0
So, we must have either
3s-4 = 0
or 2s-3 = 0
So, s = 4/3 or 3/2
Had you tried your guesses in the original equation, you would have seen they did not satisfy it.
Sure! I can help you with solving equations by factoring. Here's how you can go about solving each problem:
1. To solve the equation 4x^2 = 9, we need to get the equation in the form of (expression) = 0.
Start by subtracting 9 from both sides of the equation to get 4x^2 - 9 = 0.
This equation can be factored as a difference of squares: (2x - 3)(2x + 3) = 0.
Now, set each factor equal to zero and solve for x:
2x - 3 = 0 or 2x + 3 = 0.
Solve each equation to find the values of x:
For 2x - 3 = 0, add 3 to both sides and divide by 2 to get x = 3/2.
For 2x + 3 = 0, subtract 3 from both sides and divide by 2 to get x = -3/2.
Therefore, the solutions to the equation 4x^2 = 9 are x = 3/2 and x = -3/2.
2. To solve the equation M^2 - 36 = 16M, again, we need to make it equal to 0.
Rearrange the equation to get M^2 - 16M - 36 = 0.
Now, let's factor the quadratic expression:
(M - 18)(M + 2) = 0.
Set each factor equal to zero and solve for M:
M - 18 = 0 or M + 2 = 0.
Solve each equation to find the values of M:
For M - 18 = 0, add 18 to both sides to get M = 18.
For M + 2 = 0, subtract 2 from both sides to get M = -2.
Hence, the solutions to the equation M^2 - 36 = 16M are M = 18 and M = -2.
3. To solve the equation R^2 + 9 = 10R, rearrange it to get R^2 - 10R + 9 = 0.
Factor the quadratic expression: (R - 1)(R - 9) = 0.
Set each factor equal to zero and solve for R:
R - 1 = 0 or R - 9 = 0.
Solve each equation to find the solutions:
For R - 1 = 0, add 1 to both sides to get R = 1.
For R - 9 = 0, add 9 to both sides to get R = 9.
So, the solutions to the equation R^2 + 9 = 10R are R = 1 and R = 9.
4. To solve the equation 6s^2 = 17s - 12, rewrite it as 6s^2 - 17s + 12 = 0.
Now, let's factor the quadratic expression:
(2s - 3)(3s - 4) = 0.
Setting each factor equal to zero, we find:
2s - 3 = 0 or 3s - 4 = 0.
Solve each equation to find the values of s:
For 2s - 3 = 0, add 3 to both sides and divide by 2 to get s = 3/2.
For 3s - 4 = 0, add 4 to both sides and divide by 3 to get s = 4/3.
Thus, the solutions to the equation 6s^2 = 17s - 12 are s = 3/2 and s = 4/3.
I hope this helps you solve these equations by factoring!