Solving By Factoring
Help me with these problems.
2x(2x+1)=2
5x^2+34x=7
Please show work I want to learn!!!
4 x^2 + 2 x - 2 = 0
2 ( 2 x^2 + x - 1) = 0
(2x-1)(x+1) = 0
x = 1/2 or -1
5 x^2 + 34 x - 7 = 0
(5x-1)(x+7) = 0
etc
thanks
Sure! I'd be happy to help you with these problems and show you the steps to solve them.
Problem 1: 2x(2x+1) = 2
To solve this equation by factoring, we need to set it equal to zero first. So, let's subtract 2 from both sides of the equation:
2x(2x+1) - 2 = 0
Now, let's simplify the equation by distributing 2x into (2x+1):
4x^2 + 2x - 2 = 0
We want to factor this quadratic equation, which means we need to find two binomials that, when multiplied, give us the original expression.
To factor this quadratic equation, we'll look for two numbers whose product is -8 (1st term * constant term) and whose sum is 2x (coefficient of the middle term). In this case, the numbers are 4 and -2:
4x^2 + 4x - 6x - 2 = 0
Now, we can group the terms and factor by grouping:
(4x^2 + 4x) + (-6x - 2) = 0
4x(x + 1) - 2(3x + 1) = 0
Now, we can factor out the common terms:
4x(x + 1) - 2(3x + 1) = 0
(4x - 2)(x + 1) = 0
To find the values of x, we can set each factor equal to zero:
4x - 2 = 0 or x + 1 = 0
Solving for x in each equation:
4x = 2 or x = -1
Dividing both sides of the first equation by 4:
x = 1/2
Therefore, the solutions to the equation 2x(2x+1) = 2 are x = 1/2 and x = -1.
Problem 2: 5x^2 + 34x = 7
Let's set this equation equal to zero by subtracting 7 from both sides:
5x^2 + 34x - 7 = 0
We need to factor this quadratic equation, so we'll look for two numbers whose product is -35 (1st term * constant term) and whose sum is 34x (coefficient of the middle term). In this case, the numbers are 35 and -1:
5x^2 + 35x - x - 7 = 0
Now, we can group the terms and factor by grouping:
(5x^2 + 35x) + (-x - 7) = 0
5x(x + 7) - 1(x + 7) = 0
Now, we can factor out the common terms:
5x(x + 7) - 1(x + 7) = 0
(5x - 1)(x + 7) = 0
To find the values of x, we can set each factor equal to zero:
5x - 1 = 0 or x + 7 = 0
Solving for x in each equation:
5x = 1 or x = -7
Dividing both sides of the first equation by 5:
x = 1/5
Therefore, the solutions to the equation 5x^2 + 34x = 7 are x = 1/5 and x = -7.
Remember to always check if the solutions obtained are valid by substituting them back into the original equation to confirm they satisfy the given equation.